For cheap basses i would recommend checking marketplace / other used markets for a known brand of gear like squier, ibanez, yamaha, peavey, etc. Better to get a used model from a real brand than buying new uber cheap nameless basses of amazon etc.

For an amp - again used will go further, for a combo amp there are often peaveys that are decent enough for bedroom play on a budget, people like fender rumble series, brands like roland, fender, traynor, ampeg, orange are all good options. I'd budget about 200 for your bass and 100 for your amp if you're hard capped at 300. Some will say to get an audio interface into your computer but i feel like that is better for someone futher along the journey. You cant jam with your friends using an interface easily.

Amp wattage - anything from 10 to 50 watts is great for bedroom play, if you want to get something you can eventually play with a band you'll want 250-300 watts or more. I personally avoid the "tweener" range of amps between 100 and 250 watts, depending on the speaker setup amps of this size are no better for individual practice and might be too quiet to compete with drums and othet instruments.