It really depends on the country you are living in and which currency you intend to sell.
The OP has clearly written that he lives in the US (post number 3). Other members have posted links to more or less all the relevant guidelines that can help with the decision where and how to buy. Now it's just a matter of choice, or whether the OP wants to identify with its data and become a possible taxpayer in the future (in case of sale or payment with crypto), or wants to stay under the radar and use crypto a bit in the gray zone.
Does it take a few days to deposit a couple thousand dollars or can an of these exchanges take deposits within a few minutes?
Considering Covid-19 in the USA, I know that the verification process, at least as far as Coinbase is concerned, has been quite slow because their employees work from home. I can't tell you what the situation is at the moment, but I don't think things have fully returned to normal yet.
Here is one bad experience from some 1+ months ago
The WORST support ever. Meet Coinbase. As for fiat deposits, if you buy with debit card it should be instantly, but for bank transfers it can take 3-5 business days. You can read more info
How long does a purchase or deposit take to complete?