The Ibanez SRX line is very good for the money. I had the SRX 300 which was a perfectly giggable bass. Great sounds and a very comfortable neck to play on. You don't need to do any mods or upgrades to improve it. The hardware, pickups and electronis are fine as they are. Fret work is excellent and they can be set up for low action easily if you want that.
At a bit lower price the GSR 200 is very good too. That's the one I'd get for a PJ type sound.
I tried a couple low cost Squier basses yesterday. They were pretty good with a decent sound. I didn't like the necks all that much though. Not nearly as comfortable as the Ibanez and the fretwork is not as good. This is the type of bass where you would probably want to do a bit of work on leveling frets and smoothing off the slightly sharp ends. I don't know how the electronisc stand up with hard use. I've heard from others that they had to replace the volume and tone controls as they can quickly get noisy. The hardware seems adequate.
Lastly, if you don't mind doing some assembly and finishing work, consider getting a Bolt kit bass from Carvin. The complete package is about $420 currently for the passive 4 string model. Quality of fit. finish and parts is equal to any $1000 and more USA made bass. Their necks are unbelievably good. Wonderful fretwork. The pickups give you a very good modern Fender J type sound. They're easy to put together and finishing can be as simple as a couple coats of clear tung oil for protection.
I built one about 5 years ago and couldn't be happier with it. it's a blast to play on and very light-about 7 lbs.
Anyhow, get out and play some of these suggestions from myself and others. Then if you want to buy on line you'll be better informed to make a good decision.