Jeremy - Most all of the 3D printers will use some sort of memory stick or SD card. There are some that will connect like a printer to a host computer, but that is less common. If something happens to that connection or the computer shits out or goes to sleep it will kill your print. Never fun when you have a lot of hours into a print.
For computer you can use anything. Mac, PC, probably even a cell phone, as long as it can run some slicing software and right to the correct memory device for your printer.
Printer choices -
I'd suggest starting off with an FDM (Fused deposition modeling) printer that uses rolls of filament to print your parts. This is pretty straight forward and has the most uses for the hobby. They are cheap and generally easy to use and maintain. These can be used to print entire planes if you like, or pretty strong structures. A good one with good filament and proper calibration can produce really clean parts that can range from light but brittle to heavy and strong.
Nate's new toy is an SLA style that uses resin that is cured with a laser. He can talk more about this tool, but it has some really cool features and some real drawbacks. If you need super clean looking little things, this is the tool, but probably not much it will do for structures you may want for the hobby.
There is a vast number of options out there! I'd suggest going with a known brand that is rather popular as you will be able to get parts and assistance much better than an unknown unit. I started off with the Ender 3, this can be had for a little bit of money and will teach you a bunch as they usually have some challenge to getting a good print from them. Calibration is key. I think the majority of things I printed on my ender were upgrades for my Ender. Eventually I switched over to the Prusa brand and have been very impressed with how well they work, and how reliable they have been. The big difference with the Prusa printers is the quality of their hardware and the quality of their software. The company is always making improvements to firmwares and softwares, and most importantly, they have a ton of calibration work done for you. This pretty much guarantees great prints every time.
Don't worry about size of the first printer. Everyone seems to get caught up on wanting a huge printer for their first printer. The majority of things you will print when learning will be small, probably really small. Eventually you will want to print something bigger, and chances are that will fit just fine in a smaller machine. We have a pair of the Prusa Mini (200mm cube print area) and they can do a huge amount of the jobs we want to print. I have a larger Prusa that we had to buy for the bigger volume as we have a larger project going on in house. I like the smaller machines as they cost less, take up less space, and generally have better bed heating. All good stuff. If you know you want to print some huge thing pretty soon, and it has to be done in one piece, then by all means get the bigger machine, otherwise, get the smaller one. (I suggest minimum of 180mm build cube.)
Printing can be simple, but more often will require a bunch of messing around in CAD and slicer settings to get things just the way you need them to be. This can be a huge bunch of learning, but it is all doable. It may become a full time hobby at first, but this winds down as you learn the tools. If the project does not need to meet some higher level of quality, then printing is pretty darn easy as long as you have the basics down.
Just stick with PLA filament at first. Filament quality is important, don't use the cheapest brand you find, you will hate it. You want something with good tolerances to avoid issues with the extrusion rates, or worse jamming. Don't worry at all about all of the exostic materials or carbon or wood stuff. Stick with simple PLA while you are learning. There is a ton of stuff you can make with PLA, some people never use anything else. The vast majority of what we use here is PLA. It is cheap, it works great and it prints great. Other materials will offer better heat range, or better strength, but it ALWAYS is a tradeoff.
Hope this helps.