Italian and Chinese are great places to start.
For Italian, an easy thing to start with would be lasagna.
You can make all the parts individually, and then just layer them and bake. Each layer can pretty much be as simple, or as complex, as you want it to be.
Lasagna at it's most basic is just red sauce, ricotta cheese, lasagna noodles, and mozzarella. That's it.
Cook the lasagna noodles as instructed on the box, and set aside. I definitely recommend adding some oil to the water when you cook them, by the way, as it'll keep them from sticking to each other, when you've set them aside.
If you feel like making your own noodles from scratch, more power to you. I suck at it, personally.
Layer noodles, ricotta cheese, sauce, mozzarella, and repeat.
The fun (as it is with most recipes I really enjoy) is in adding things, and making the basic stuff more complex.
It makes it a creative act.
I like to add (as separate layers) cooked Italian sausage (cased, cut into inch long pieces), pieces of chicken, beef, and pork (again, each meat in a separate layer), which have been breaded and fried separately, peeled and sliced eggplant, zucchini, whole mushrooms (separate layers), and even fresh baby spinach.
A good version is spinach and Italian sausage.
To the redsauce I'll add burgundy wine. The wine thins it out, and also adds a lot of flavor, and some acidity. I can make a passable redsauce from scratch, but I don't have a big problem with the stuff that comes in a jar, when it comes to lasagna, as long as it's of decent quality. The best thing about the jar stuff is knowing beforehand exactly what the consistency is going to be. I find that's important for this dish.
In addition to redsauce, a good Bolognese is also a possibility, but it's pretty advanced, and because of the complexity of Bolognese, it's harder to add to it, profitably. I really am not that fond of it in lasagna, but it's a great sauce. I'll write up a recipe for it sometime, though. Maybe next time I make it.
To the ricotta cheese, I add chopped herbs (fresh basil and parsley, if I have them. I rarely use much oregano, except in hummus, because it upsets my stomach, but your mileage may vary.), melted butter, crushed garlic, egg yolk, and a little sugar. The sugar helps the ricotta cheese stand out, while the egg helps it "set up" while baking.
In addition to mozzarella (but not replacing it), you could also use provalone, asiago, and fresh parmezan. I'd suggest no more than 1 part other types of cheese to 2 parts mozzarella.
If you're going to use romano (and I love romano), I'd suggest just putting it on top, because it's got a lot of salt, which might throw off some of the internal flavors going on.
When baking, 350 degrees (Farenheit) for 45 minutes for 1 box of noodles is about right. It might need an hour, depending on the thickness. You want the cheese on top to start getting brown--part of the purpose of the cheese on top, by the way, is keeping the sauce from overflowing.
It's easy to fill the baking dish too full, so try to leave a little room on the top, to prevent the lasagna overflowing while it cooks.
If it's *really* thick, you could maybe reduce the heat to 275 degrees, and cook it for an hour, then raise the heat up to 350, and cook it for another 15-20 minutes.