As mentioned, as a Geddy fan you should definitely check out a Squier Jazz bass, they're very good basses for the $300 you'll spend. And spend another $100 on a low wattage Fender or Peavey (15w to 25w is perfect) - these are durable, sound decent, won't break the bank or annoy your neighbors/roommate/significant other

If you decide to play with others at some point, you can always get a bigger amp and keep the smaller one to port around the house easily or play with someone playing acoustic guitar.
I will also recommend the Ibanez SR300. These are also around the same price as the Squier and they're very different from the Jazz bass in that they're smaller bodied, more contoured, have thinner necks and active electronics as opposed to passive on the Jazz.
As far as electronics, what that means is the sound is more tweakable on the Ibanez. This does not equate to BETTER, just different.
As far as everything else, the SR300 will arguably be more comfortable to hold and play since it's lighter, smaller, thinner. But the Jazz is a time tested design so it's not like you'll likely have issues with it. I'm just recommending the SR300 because like the Squier Jazz, it's an excellent buy. So with a $300 budget, these are the two you should be checking out.
Now if you're buying used, you should be able to find these two for around $200 and for up to $300 you can also find a bunch of other good buys such as a Yamaha BB400, USA made Peaveys that have been mentioned, and real Fender Jazz Basses (Squier is still Fender but it's a more affordable sub-brand, the lower priced Fenders are made in Mexico, the higher priced version made in the USA).
Good luck! You'll have a blast with your new hobby!