In the el-cheapo range, the choice is pretty much like that of the guitar world.
Its a case of sitting down, working out your budget, and whether you want 4 or 5 strings, about ?300 will get you a bloody good bass, that'll last for ages. Then its a case of going to your local music shops and trying a few out in that price range. There are some great bargains to be had in the second hand market of late 80's basses (like Aria Pro II's), and the new stuff you have the choice of loads, like Squier, OLP (MusicMan copies) and a load of others (mainly built in Korea or China). Try before you buy, just like a guitar.
Your main choice of the Fender style would be either the Precision or Jazz style of copy. The P tends to have a chunky neck, where the J has a thinner neck. Both are long scale, and may take a bit of getting used to.
In the Gibson style, your choice is a little more limited, but things like Epiphone EB3's (SG with 2 p'ups) and Rivoli's (335) tend to be reasonable(ish) and of a shorter scale.
Don't forget, even though you have 2 less strings, the gauge is bigger, and so is the cost. I like Rotosound RS66LC (40 - 95) which are cheap (about ?15), but you can pay in excess of ?20 for a set.
If you play an LP, with a thick 59 style neck, I'd recommend something like a P bass, as the neck will feel comfortable.
I did it the other way around, went from Bass to guitar, and so an LP 59 style neck feels nice to me.