The best pizza sauce uses NO paste. Paste is too sweet.
1) Chop onion and garlic. I like about 1/2 cup chopped onion and 1 tbsp chopped garlic.
2) Simmer the onion and garlic in olive oil. Cook ~10-15 minutes for mild taste, or ~5 min for sharper garlic and onion taste. For even more flavor, only cook half of them and toss in the other half right at the end (uncooked).
3) Finely chop, or mash, or blend one large tomato (or two medium tomatoes).
4) Add the tomatoes to the garlic/onion when the desired cooking time has been reached.
5) Add a small can of tomato SAUCE. You could use more of your fresh tomatoes instead of sauce, but it'll need to simmer longer.
6) Generously add basil and oregano. Try at least 1 tbsp each. I'd do it in stages and taste it until you like it.
7) Generously add salt until it isn't sweet any more.
8) Add a dash of black pepper and some red pepper (flakes or ground). I like a lot of red pepper (1 tsp at least), but I like my food hot.
9) Simmer until thickened. The more you simmer, the better it'll be. But it is fine after 5 minutes if you are in a rush.
The whole thing is about 5 minutes of actual labor and can be done in as little as 15 minutes if you are in a hurry. I really like to let it simmer an hour while I make the pizza dough and toppings though.
Change the spices in step 6 and you have a chili or spaghetti sauce base. Just add beans, meat, green peppers, more onion, and cumin for chili. For spaghetti, most people will like it with mushrooms or eggplant or green peppers. This is really a versitile sauce, and it takes so little time to make. It freezes quite well too, so you can make it now before they rot and eat it months later.
It is better with roma tomatoes since romas need less simmering time. But the Jersey tomatoes will work just as well.