1) What would you recommend I use to store my BTC? I was thinking of looking for a program I can just run on my computer to store my crypto.
Ideally, you should use a non custodial wallet to store your cryptocurrency. This type of wallets gives you alone access to your funds as you alone can access your private keys, as opposed to custodial services like coinbase which can lock your funds. You can run Bitcoin core on your computer and generate your own address.
2) Do these wallet programs allow you to buy and sell BTC?
Custodial wallets are only for storage and transactions but does not allow one to buy or sell bitcoins.
3) If I use a hardware wallet, cold wallet etc, how do I go about cashing out crypto? Do I have to use an Crypto Exchange? Are the fee's going to be similar no matter what I use?
You can trade OTC or directly with a dealer, there are some of them in the currency exchange board on the forum (be careful to check their trust score and previous trade history).
For more liquidity, you can also use an exchange to trade, but ensure you do not store fund there.