Welcome aboard, David.
The 1624-44 is a good lathe. I've not used one, but I know a lot of guys who like theirs a lot.
The "what else do I need" question is never fully answered...you'll find the lathe purchase is just the beginning of your fall into the vortex. (It's a fun fall, though.) Before you get too carried away buying accessories and gadgets, I'd recommend going to the club meeting and seeing if there are any guys with some spare time to give you a little mentoring. That can shorten your learning curve a lot, and save you quite a bit of money, too.
One thing I WILL recommend is a good sharpening jig to go with your new grinder wheel. Sharp tools are a must, and without them, turning is not very much fun at all. A jig makes it easier and faster to get the tools sharp, and you'll grind away less of the tool each time by using the gig. The Wolverine by Oneway is the most popular sharpening jig, and for good reason...it works well. A lot of folks have also made their own version of the Wolverine style sharpening jig. They also work, but if you have the funds, I'd recommend just going with the original.
You'll also need some gouges and scrapers and skews. A lot of us started out with the HSS set from Harbor Freight. They are not great tools, but for their price, they are a pretty good starter set. The next step up quality-wise would be something like the Benjamin's Best tools from Craft Supplies USA. (The Wood River tools from Woodcraft are comparable, but I think they're a bit more expensive.) They are somewhat better than the Harbor Freight tools. The next step up from there is getting into what I consider REAL turning tools, and at that point I'd recommend against buying set, and instead just buying the individual tools you need. (Experience will tell you which tools those will be for your needs.) There are several well-known and popular brands like Sorby and Crown, but for
less money, you can have the best: Thompson Tools. (Personal opinion: Sorby and Crown tools are priced at about 2 to 3 times what they're really worth. They are good tools, but priced higher than great tools.)
And don't forget the safety gear. A face shield is a must. Not just goggles...full face coverage, in my opinion. At some point in your turning adventures you WILL get hit in the face with stuff, Better to have the shield on before that happens. You should also have some lung protection for when you're sanding...a respirator, a dust collector with a good lathe scoop, or minimally a good fan blowing the dust out your open shop door.
Welcome to the Abyss!