I purchased $300 worth of BTC on coinbase, should I keep it on coinbase or move it to a wallet such as exodus or blockchain?
I mean if you are fine to pay the notorious fees, then you should move it to another wallet. Blockchain is a better choice than Coinbase.
Having a hardware wallet is the safest option.
Even though I don't recommend keeping BTC inside exchanges you'll be paying very high fees every time you move your BTC and at this point wouldn't recommend it unless you really have to.
move to blockchain ,
Enable 2 factor auth and you should be safe.. 300$ worth of BTC is nothing.