Longtime lurker originally from BR. Moved out of state but still check-in regularly. I work for an acclaimed specialty coffee roaster, and will try to answer any and all questions you may have. From roasting to consumption, let me know if I can help you make your coffee experience better.

Ever heard of Titan Coffee?

Nola has some great local roasters. Amazing, the polar opposite in Jefferson Parish which is a wasteland for the same.

New Orleans is definitely going to be a contender with more known specialty scenes in a few years (Portland, Seattle, Phoenix cough cough...)

Stay away from espresso in a home setting.
Get a Baratza Encore grinder, a decent scale, and a Chemex. Learn to use them! It takes some time and effort, but well worth it. Also a gooseneck electric kettle is a game changer.
You can, however, make just about anything work. My favorite method: get a french press, add 32g coffee, bloom with 75 grams water (just off boiling), wait 30 seconds and add water to total 575 grams (375 if you want it iced). Wait an additional 3 minutes and 30 seconds, break the crust and scoop off the orange foam/fat layer. Put the lid/plunger on but leave the plunger an inch above the coffee. Pour against the metal screen into a paper filter to remove all particles. It really cleans it up! If making iced, add 200 grams and enjoy.
Starting points for home roasting are a Gene Cafe or Behmor 1600. But you can have a ton of fun with a popcorn popper (with side vents at the bottom) or just a skillet and a wooden spoon. But make sure to go outside! Smoke alarms will go off, and the house will smell like burnt bread for days.

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Generally, I am a pussy when it comes to coffee. It burns my mouth, and I generally can't drink it without more sugar and cream than coffee itself.

If I wanted to become a real coffee drinker, for reasons, where should I start? What coffee? What brew method?

My "aha moment" was when I had my first full natural Ethiopian done right. If you like fruit bomb cabs, this is pretty much the same experience—huge fruit notes.
Drip coffee makers tend to pull a ton of acidity out of the coffee when brewing. Better methods are chemex, cold brew, or aeropress. None of these are too costly, and cold brew can be made with equipment you may already have on hand. Best method for cold brew: 10 ounces of water by volume to 1 ounce of coffee by weight. So if you have 16 ounces of coffee (ground coarse), you'd need 160 ounces of water. Add coffee and water to a vessel (jar, pitcher, whatever), stir, cover, and let sit on your counter top for 20 hours. Strain through a filter, chill and enjoy. Finished product will stay good in your fridge for up to 30 days depending on how clean your equipment was.
When coffee is done right, you shouldn't need to add anything to it to make it drinkable.

Age is everything. I moved to buying green beans and roasting at home. It was much more noticeable when I moved to espresso vs French press. The rancid stale oil flavor after a week or two was very forward. I’ve saved a ton of money. Green beans run me about $5/lb and I get to sample single origins and blends

If it's for home use: Huky, Hottop, and Toper make great smaller roasters, I've used most of their 1kg models for sample roasting. It can be harder to control heat, ROR, and consistency overall when working on a smaller roaster.
If you are wanting to get into commercial, you might as well just go ahead and buy a 12kg. Probat is the only thing I will touch, the P12 and P25 are game changers.

If you can find a Starbucks with a Clover brewer it's the closest thing to a decent cup they have. Try one of their reserve coffees, I used to drink them all the time. It will still taste burnt, however.
I have to also say that without Starbucks, the specialty coffee scene wouldn't exist. They got people to pay $5 for a cup. They were second wave, specialty is third wave.