You're off to a great start! I think the Cub is the best tractor out there for this sort of thing. It's got the belly mount and offset engine, and it can have a rear lift too. A couple years ago I grew 4 acres for market, mostly on plastic mulch but some with the Cub. This season I'm growing two acres for wholesale, all with the cub and cultivation implements of my own design. The cub's got all the capability to do everything I want, though I find it's not perfect "out of the box" for a modern vegetable farmer, since it was designed for the old-school commodity farmer. Changing implements takes undoing lots of bolts, and that's slow and annoying. I'm making rear and belly 3-pt hitches with modern pins to get around that. And the 144 cults are meant for row-crops, a single row down the middle, with the two inner standards adjusting in+out based on what's good for your corn etc at that point in the season, not for vegetable beds.
Basically, you probably want a belly-mounted toolbar, and attach whatever tools you want to it for your 3-row system, you could make up a toolbar for that pretty easily. Track sweeps in the rear to keep the aisles clean.
What tools you want depends on how detailed a system you want to have, and want to pay attention to, and what your soil's like, and how much labor for hand-weeding you have. Most basic would be 4 sweeps (solid standards or c-shanks) on one toolbar for a 3-row bed. Less fiddling, less cost, and less effective than fancier tools -- but maybe smart to go simple the first year and then see what problems you have to fix for later with a more detailed system. Whatever you do, do it consistently and often, and don't miss your chance when the soil's perfect and crumbly after a rain -- at least, my soil gets crusty if left too long in the hot sun, and then it shatters rather than crumbles.
The trouble is getting close to the plants; when they're small you're in danger of burying them with sweeps, and then weeds grow in the uncultivated strip beside the seedlings. I'm a fan of the basket weeder (and I made one of those too, because Buddingh's are super expensive) -- it cultivates shallow and doesn't throw much soil, so you can get closer to small plants. But then the basket weeder gets too wide for large plants. If I had just one tool though I'd take a basket weeder.
Stalebedding/bare fallow is a HUGE help, it kills weeds before you plant the crop. A tine weeder or basket weeder setup for that, or even in a pinch light discs or a field cultivator run shallow, but you really don't want to disturb much soil and bring up more weed seeds.
I'm planning--
shape bed
stalebed for a few weeks
mark beds with the cub
seed with a push seeder (Jang)
really soon, basket weeder for between rows with some light tools for getting as close as possible to plants
basket weeder, for between rows
(maybe tine weeding, for getting in-row weeds, once the crop is tall enough to stand it)
sweeps, since the standards can fit down between the crop canopy, and the sweeps do some hilling to bury weeds next to tall crop plants
Rear track sweeps as necessary or when conditions are good.
Lots of implement changes, so, modern hitches are a super help!
Have something in mind to cultivate the sloped edges of the raised beds, and the wheeltracks. I like c-shanks or s-tines for the wheeltracks, they get into my firmer soil better and shatter it, where the sweeps (like on spring-trip standards) just dig up chunks. But if you have lighter soil sweeps would probably be great.
I haven't found a good source for new standards and things, best probably is keeping an eye out on the for-sale section here. Or tractor yards nearby, or auctions -- I just got 20 standards at an auction off a 4-row row-crop cultivator, which will hold me for a while!
I apologize if this is too basic and you know it already -- but a good starting point I suppose. Feel free to PM me if you want more details, I don't know what's useful to you.
-David