Great! I'll go into a bit more detail then so you don't have to start from scratch.
I know the stock market is booming right now and I'd like to take advantage of that before we hit another recession.
Lets address this first. If you're going into a tax-advantaged retirement account, chances are you're not going to touch the money or earnings for 25 years. You're likely going to see multiple booming markets and recessions. Recessions are actually decent times to jump into the stock market, since you can get more per dollar spent. However the most important thing to remember is that you cannot time the market. You can't predict when it will peak, you can't predict when will be the best time to purchase, you can't predict when will be the best time to sell. As a rule of thumb: Time in the market beats timing the market.
Now, the two big retirement investment accounts are IRAs and 401ks. 401ks allow you to contribute pre-tax dollars, while the capital gains will be taxed. 401ks are offered through your employer (though there are some oddball options out there), sometimes the employer will match you contribution up to a certain point (which you should take full advantage of if available). Sounds like your employer may offer one through fidelity but could be only for full-time employees. You don't have as much freedom as you'd have in an IRA.
IRAs allow you to contribute post-tax dollars (your actual take-home pay) and grow that money tax-free. You can contribute up to $5,500 a year (January-April of next year), and you can pick what to buy. You don't have to rely on your employer for anything but income in this case. Unless you have an employer matching 401k, this is what I'm expecting you to do.
Your original question is about brokerages. Picking the right one is fairly important, though switching over shouldn't be a major hassle if you don't like where you're at. You want to try to give your brokerage as little money as possible, so when you're picking you need to consider the following: What do they sell and how much will it cost me to have it?
I only buy low cost index funds with no commission. Vanguard used to be king in this area, but other companies are starting to catch on, like Schwab. You won't go wrong with either. If you look elsewhere, just be sure to see what commission free ETFs they have available. I don't use either and can still buy the bread and butter of my portfolio (VTI and VEU) without paying commission.