The word rustic was much used during this class. And it probably applies as much to the tomato sauce as to the pasta.

There seems to be some not insignificant cleavage between Italian-American tomato pasta sauces, and the kinds originally made in Italy.
Salsa is the Italian word for any kind of sauce, including a dip, whereas sugo is the Italian word generally used for cooked tomato sauce, as might be served over pasta.
Sugo al pomodoro (Tomato Sauce) in Italy is almost always a fairly quickly-cooked sauce made with fresh tomatoes. Usually garlic and optionally a vegetable soffritto is fried, then chopped fresh tomatoes are added and reduced to a purée over 30 minutes to one hour. Herbs being added at the end.
The only concession to preserved tomatoes made in Elizabeth David's iconic cookbook Italian Food is the addition of tomato concentrate to the end-of-season tomatoes in her Salsa di Pomidoro (3).
There are no tinned tomatoes in sight.

In Naples they make a similar but even quicker tomato sauce called sugo alla marinara, so-called allegedly! because the Neopolitan fish-wives would put the sauce on to cook when they caught sight of their husband's fishing boats returning to port, and have dinner on the table 10 minutes later when he stomped through the door with his sack of anchovies.
This name is virtually unheard of outside Naples, except as a pizza topping, but is very common in America.

Then there is ragù, which is a longer-braised stew invariably containing meat. Bologna's ragù being the archetype. Tomato-less ragùs are also a thing.

Tomato sauces in the U.S. in contrast, are more often cooked down for many hours. Indeed the length of time your Nonna's Restaurant cooks her sauce seems to have become a competitive sport amongst Italian migrants. This may be due to the fact that these are rarely made from fresh tomatoes, but almost always tinned varieties. Probably because access to fat, ripe, glossy tomatoes was extremely limited for a family of Sicilians newly arrived in East Harlem's Little Italy.
Despite the irony of tomatoes being originally imported into Italy from the Americas.

Most Italians will press their cooked tomato and vegetable sauce through a passatutto (food mill) but since we didn't have access to those in class we just used a hand-blender and a coarse chinois (conical sieve). The result is probably an inferior passata (the product, unsurprisingly, of the passatuto) which doesn't contain the crushed seeds, stalks and foamy air introduced by blending.
But hey - we work with what we have.

Teacher Paul's approach to making his tomato sauce is something of an Italian / American hybrid. We used fresh tomatoes, but cooked them very slowly for several hours. Or would have, if we had the time. Teacher Paul recommends simmering with the surface barely moving - he describes it as warming rather than cooking. Or sitting them overnight in the bottom of a very low oven.

Once you have your basic sugo al pomodoro you can expand it in various directions:
Throw in olives, capers and anchovies and you'll have puttanesca.
If you add chilli you get an angry sugo all'arrabiata. If you continue with guanciale, pancetta or (spit!) bacon you can make bucatini all'Amatriciana, and if you also throw in some eggs and Italian sausagemeat you can end up in a filthy mess with pasta alla zozzona.
The world of tomato-based pasta sauces is now your oyster. (But not your clam - linguine allla vongole would not usually involve tomatoes.)

And so... on to the rustic pasta.
One of the important lessons Teacher Paul transmitted is the value of using a coarse semolina when working the prepared pasta dough.
Generously scatter semolina over the pasta and work surface when you roll it, fold it, and cut it which will help prevent the dough from sticking whilst simultaneously keeping it dry and avoiding it becoming over-floured.

When it comes to turning your thin translucent sheet of rolled dough into flat pasta ribbons for boiling, you have a series of width and thickness options.
Technically neither spaghetti, which is round, nor linguine which has an oval cross-section are available to you. So:

• tagliolini or tagliarini also known as tajarin in the local dialect is a very narrow (2-3mm - half the width of linguine) ribbon from the Piedmont area in the North West of Italy, capital Turin. Best served with light delicate sauces.
• fettucine (5-7mm) is a wider and thicker ribbon from the Lazio area around Rome. Typified by fettucine alfredo
• tagliatelle (6-8mm) is usually slightly wider and thinner than fettucine and hails from the North-Eastern Emilia-Romagna region, especially Bologna with whose rich meat sauce ragù alla Bolognese it is famously associated. Not spaghetti - despite what you might have heard in Britain.
  Apocryphally it was styled after Lucretia Borgia's hair. Who must have looked like a badly animated cartoon character if so.
• pappardelle is the widest of the ribbon pastas at up to 1" (2-3cm). It comes from the central Tuscany region, is sometimes given scalloped or fluted edges in modern formulations, and is frequently paired with mushrooms or robust game stews.

Paul'sRusticPointers™

• Fresh pasta freezes well - just sprinkle with flour or semolina, roll up into nests, partially freeze until stiff on a tray, then seal into freezer bags and freeze completely.
  To cook - drop directly into boiling water without thawing.
• You can freeze tomatoes whole, like snooker balls, for adding to sauces or stews if tomato plants have completely taken over your garden.
• Teacher Paul's tomatoes grow better in his garden than in his greenhouse. I'm not sure how this is relevant.
• Fresh pasta will keep in the fridge for a couple of days, though it may discolour. Be sure to sprinkle generously with flour or semolina and roll up into nests so it doesn't stick together in a messy clump.
• Teacher Paul recommends freezing the tomato sauce in ice cube trays so you can just pop them out to use individually when making sauces. Though I personally would need to eat about 20 of them at a time, so maybe larger containers?
• Pasta is best made with a strong gluten, high protein content flour like that made from the durum wheat variety.