I would ignore the current trend for 5 or 6-string basses. Again, these have wide necks. You shouldn't consider a fretless until you have been playing for a few years.
Fender copies still abound, and you can pick up decent new ones for £150/$250. These are safe instruments, but personally I find Fenders and therefore Fender copies ugly. Fender's Jazz may be easy to play, but their Precision has a very wide neck which might deter some beginners. I reckon you get a better deal with original instruments from the makers Yamaha, Ibanez, Washburn and Aria. Tanglewood are currently a popular budget make whose basses sell well here in the UK. A few years ago I bought an attractive budget model from Peavey, the Zephyr, selling here for £250/$420 which has the bonus of a through neck!
I would ignore the current trend for 5 or 6-string basses. Again, these have wide necks. You shouldn't consider a fretless until you have been playing for a few years.
There are several extras to consider when you take up the bass. There's no point in getting an amp until you start playing with other musicians, but will you need a case for the instrument? A tuner of some sort? A strap? A stand or wall-hanger?
By the way I always fit Schaller locking strap buttons to my basses. These are really secure, and you can swing your bass round your head safe in the knowledge that it won't come off the strap.
The bridge itself should have some kind of tracks or grooves to stop the saddles sliding sideways under heavy playing - a feature sadly lacking on many Fenders. My own preference is for bridges which allow full 3-dimensional adjustment. That way you can set the gaps between adjacent strings. I sometimes find when playing that the G-string occasionally slips off the edge of the frets, and bringing it in a bit certainly cures that. Schaller and Badass both make excellent such bridges. These feel heavy in your hand and can be bought as upgrades.
Most basses have bolt-on necks, which does mean you can replace them should you be unfortunate enough to break one. If your bass has one, periodically check the bolts are completely tight. My own preference is for a straight-through neck, where both ends of the strings are attached to the same piece of wood and the body of the guitar is stuck on at either side. Such basses though tend to be more expensive and are pretty rare in the shops over here these days. A third kind is the glued-in neck, every bit as strong, which I think some Gibsons and Rickenbackers have.
Some of the more upmarket basses feature “active” circuitry. An on-board battery powers a little circuit which can boost various frequencies. Don't pay over the odds for this - it's a bit of a luxury.
Machine heads should turn smoothly, and strings should be of the “roundwound” variety. If your bass has archane “flatwound” strings with a layer of flat metal tape wound round them then bin them at once and buy a roundwound set! You will be amazed at the difference in tone.
The height of the strings is important, and is something of a compromise. It can be raised or lowered by adjusting the saddles at the bridge, usually with a small Allen key. If you raise the height of a string you will get rid of all annoying fret buzz, but go too high and it becomes much harder to play. Tone improves somewhat with height, but far more pressure is needed to press the string down and fast runs become all but impossible.
You might imagine that the ideal bass neck is perfectly straight when strung, but it's not. The lower half of the neck should have a very slight dip in it, the upper half should be virtually level. Hold the bass as if you were playing it and press a string down simultaneously at frets 1 and 10. Pluck this section and the string should ring out quietly but definitely. Now repeat between frets 10 and your highest fret. The string vibration should be damped by the frets inbetween.
If the neck is not like this, or your bass suffers from fret buzz and the strings have to be raised too high for comfort, you may have to adjust the truss rod, and even this may not always help. (If a neck is really bad, a professional refret or similar may be the only way to make it playable.)
The truss rod is a metal bar running through the core of the neck. At one end it is threaded with a nut, usually turned by a larger Allen key. You can simulate the effect of tightening this nut by holding both ends of the neck. Now gently pull the headstock 5mm or so towards you. The strings get tighter and the action falls. Push away from you to simulate loosening the truss rod.
Always be careful if adjusting the truss rod. I have heard of necks snapping due to the force generated! Adjust in small steps, say a quarter or a half-turn at a time. For larger adjustments, spread them over a few days to allow your bass to recover from the shock. Don't forget to keep strings tuned to the correct pitch when you adjust the action in any way.
A couple of other things you can adjust are the intonation and the pickup height. Intonation means the vibrating length of each string, and this is altered with the screws that run parallel to the strings at the very rear of the bridge. You will probably notice that the deeper strings are set slightly longer than the higher ones. The length is correct when the fretted note at the 12th fret is exactly the same pitch as the harmonic note produced when you just lightly touch the vibrating string directly above the 12th fret. This is best checked using an electronic tuner. If the fretted note is higher than the harmonic, lengthen the vibrating length. If lower, shorten.
Setting pickup height correctly is a surprisingly difficult task, even more so if your bass has two pickups! Your aim is to ensure all the strings sound equally loud when plucked with equal force. Each pickup can be raised or lowered by screws. But you cannot easily judge a note's loudness by ear alone, so the best way is to use some sort of VU meter or the indicators built into some fancy bass amps.
I tend to find that a properly-adjusted pickup is lower at the E-string side than the G-string side. If your bass has bridge and neck pickups, the neck one will probably need to be lower than the bridge one.
When you ask to try out a bass in a shop, the assistant will probably try and plug you in to some fancy amp. Stop them! Spend the first ten minutes playing the bass without an amp. That way you can hear fret buzz and unevenness of tone far better, and if you are a complete beginner then you won't feel embarassed at everyone in the shop hearing you! If you can't hear a damn thing because some hap is blasting out a loud guitar, ask if there's another room you can sit in.
Begin by trying out every single fret on every string. Remember to place your finger just behind the frets, not on top of them. Play each note several times, softly and loudly. Are you happy there's no bad fret buzz? Most new basses in shops are fairly well set up, but a competent assistant should be able to make small adjustments to the string height if required. Now try it all over again doing little hammer-ons (eg 7-8) or semitone trills (eg 5-6-5) with each fret and the one above it. This is to test the sustain. Pluck the string once only, and see how long it still rings out at the end of each of these. Many basses fade a bit around fret 9. Such “dead spots” are not really serious, but if you find one it might assist you in haggling the price.
Now if you like you can use the amp! Remember you are testing the bass, not the amp, so set all the amp controls to dead centre. You really only need to check that the volume and tone controls and any selector switches on your bass do their job properly.
Regularly clean the machine heads and knobs, as sweat from your fingers can cause corrosion. To clean grime from the fingerboard first remove all your strings. You then use a few dabs of T-Cut (available from any car accessory shop) on a sheet of kitchen paper. Add a smaller dab of shoe polish - dark tan for rosewood, black for darker wood - and rub the paste into the wood. Remove it promptly with clean sheets before it dries, and be sure to get all the bits right up against the frets. T-Cut dries a white colour, so the shoe polish prevents discolouration if any traces are left behind in the grain of the wood.
Now your fingerboard is nice and clean, you can treat the wood to a few drops of Almond Oil. Just rub it over, wait a couple of hours and again clean off as much as you can with paper towels. By the way some basses such as Rickenbackers have varnished finerboards, in which case you obviously don't need to do any of this!
After a while, the strings on your bass will get dirty, lose their bright tone and develop stripes where they rub against the frets. A temporary solution is to boil them for five minutes in a pan with a little detergent, watching the pot doesn't boil over. Eventually though you'll have to replace strings. Bass strings are quite expensive, but a new set always sounds wonderful.
This all dictates a fairly high strap position. It is impossible to play the bass properly when it dangles down around your abdomen, however cool it might look! I find a strap length of about 97cm (38") hole-to-hole suits me nicely for most basses.
Regarding the right hand, there are three main schools of thought. One is funky slap-and-pull playing, which was especially popular in the mid 80s. The second is using a plectrum. This is good if you want to play fast repeated notes such as in Thrash Metal.
My own personal advice would be to use the third method, fingerstyle playing, because it is much more versatile and natural. Here you simply pluck the strings using the index and middle fingers. Try and alternate them. Your fingers should come down almost vertically to pluck the strings. Keep your right wrist high above the strings, and rest your thumb on the E-string or the edge of a pick-up. Try varying your right hand fingernail length, and see what effect it has on the tone you get. Beginners will get little white blisters on these fingers, but these soon develop into hard skin and are not a problem.
You will also need to learn a bit of music theory. Initially this need only be how major and minor and maybe seventh chords are made up. If you are new to all this, then ignore the crap you get in books about thirds and fifths. I always think in semitones, so a major chord would consist of the notes 0,4,7 played one after the other where 0 is your starting (root) note.
Try this on your A-string - 0 (open string), fret 4 with your index finger, fret 7 with your pinkie. That's the three separate notes which form the chord of A major.
Now try playing the chord of C major using the C on the 8th fret of your E-string as your "0". Ask yourself, relative to that fret, where are the "4" and "7"? That's right - at frets 12 and 15, because you are starting eight frets up. So, on your E-string, 8,12,15. Hmmm, not so easy this time! Maybe we can try playing the "4" and "7" on different strings instead of at frets 12 and 15? How about this : 8th fret E-string (0) with your pinkie, 7th fret A-string (4) with your ring finger and 5th fret D-string (7) with your index finger. Same notes as 8,12,15 on the E-string, same musical spacing between them, but played in a different (easier) place.
Likewise a minor chord is 0,3,7 ; a seventh chord is 0,4,7,10; a minor seventh is 0,3,7,10 and the cool-sounding major seventh is 0,4,7,11.
There are many websites featuring lessons for beginners or tablature for many of your favourite songs, but whatever you play please practice, practice, practice...