What do you buy? Can't answer that, because it depends on where you shop and how big your budget is.
I will say this, however. The bags of stuff usually available at big box garden centers are, in general, crap. They are mostly filler (no matter what it seems to say on the bags). And you don't want Miracle Grow, because it's synthetic fertilzer is much too high in nitrogen---with the result that you get bushy plants, with lots of leaves, but comparatively little flower and fruiting.
On the other hand, nurseries usually are out of the soil & amendments business by this time of year. You might check around, though, and see if anything is still available.
Assuming you will be using the humus that you've build over the years, I would get about an equal amount of topsoil. Mix the two together (just use a hoe or a rake right in the bed). Then top it with compost, if you can find any this time of year. If not, wait until spring and do it then. The compost does not have to be worked in to the soil, but it doesn't hurt to do so. Meanwhile, you'll be adding organics to the bed between now and spring as they happen.
You'll probably have to haul the topsoil yourself (assuming you can find it). A 4 x 12 bed, 12 inches high, is only 5.3 cubic yards. In your case that amounts to about 2.75 cubic yards, which might not meet their minimums for delivery. But it will fit in a pick up. And you save the delivery cost.
If all you can find is the bagged stuff at the box stores, go with the best grade of topsoil they have. But increase your humus ratio to as much as 3:2 if you have that much. This will create a light, airy tilth.
And yes, the grass thing will work fine. Just remember to mow it close, and use the weed guard. With a new bed I usually lay down the weed guard first, then put up the walls, which helps lock the paper in place.
12" is plenty high for a raised bed. Unlike a container, you have a soil base under the bed, and that provides plenty of room for deep rooted plants. They'll punch right through the paper weedguard. Even 10" is plenty.
What have you decided to use as walls?
Just as a sidenote. I know you're anxious to get going. But do not bother with other amendments such as aged manure, bonemeal etc. at this time. Some stuff you grow will need them. But if you add them now there's a good chance they will leech out (there's that word again) over winter. Instead, once you know what sort of amendments you'll need, it's best to work them in about two weeks before you actually start planting.
Don't forget, too, that many of the things you want to plant have to be started, indoors, a long time before last frost. Tomatoes, for instance, are started 6-8 weeks ahead; peppers 8-10 (or even longer, for some varieties. Peppers can take as much as 21 days just to germinate). Brassicas (cabbage, broccoli, etc.) are planted 3-6 weeks before last frost. So you want to back off six weeks from whatever that date is and start them.
One other thought. As you research this two things will happen. One, there's a good chance of becoming intimidated by the mass of information to absorb. Don't let it throw you, it comes in time. Second, and more importantly, there is a tendency among garden writers and members of gardening forums to be very dogmatic. They will insist that such-and-such is the only way, or the "best" way to accomplish something. That is, in a word, nonsense! There are numerous ways of achieving any gardening goal. What works for me may or may not work as well for you.
My friend Roger Postley is a well known heirloom tomato specialist (growing as many as 100 varieties annually). When he gives presentations he starts by telling the audience: "There is absolutely only one way to grow tomatoes---and that's the way that works for you!" Keep that in mind when the advice seems to become overwhelming.
Here's a real life example. When I plant tomatoes I dig the hole. In the bottom goes a handful of compost. Then a tablespoon of Epsom salts and the contents of a book of paper matches, followed by another handful of compost. Then I back-fill, and finish by pouring a circle of powdered milk around the seedling, and put my cage in place. I can, if you wish, explain why I do all that. But hold that thought.
Carolyn Male (author of 100 Heirloom Tomatoes for the American Garden), whom many insist is the queen of tomato growing, says that the only things that should go in the hole is seedling and soil.
The difference is, I present my method only as one that works for me. And she insists that her way is the only correct one.