1. Specific recommendation for pots/pans and knives. Probably looking to spend around $100-150 on pots/pans and $50-75 on a couple of knives (I've heard a chef's knife + utility or paring knife is a good combo?).
2. What are some basic cooking skills that I should know (ex: knife technique, etc) and what are good resources for learning these?
Thanks!
1a. 10" or 12" nonstick skillet (frypan), T-Fal Total Professional, and a 12" clad skillet if possible (stainless steel over an aluminum core). Tramontina if you're on a budget, All-Clad if you're not. If you're on a really tight budget, get stainless steel with a decently thick aluminum base welded on, and a smallish (3 qt?) pot too. I would also advise getting a $40-$50 enameled cast iron dutch oven, 5-6 qt size.
1b. Budget knives: get a vegetable peeler, a Victorinox Forschner 3.5" paring knife, and a Victorinox Forschner 8" chef's knife, plus a bread knife (same brand) if you are going to be slicing your own bread a lot; otherwise you can postpone the latter. Also, get a ceramic honing steel from Ikea if you can, and a manual sharpener (Accusharp's isn't bad, Ikea's rebranded Fiskar sharpener isn't horrible IMO either, or you could just get a series of fine grit sandpapers).
And a cutting board, maybe two (one for meat and one for veg). The main things you want in a cutting board are (1) it doesn't damage your knives - so absolutely no glass and (2) it's durable yet doesn't trap bacteria.
Always hand-wash your knives and put them away dry. A knife block is nice, or a magnetic strip, or you can get a knife drawer tray. Worst case, make some cardboard sheaths for your knives so you can put them in a drawer without them nicking each other.
2) Slicing carrots, slicing/dicing onions, chopping garlic. Sauteeing aromatics, searing meats. Braising is great to turn more inexpensive cuts into delicious food, though I do tend to think of it as more of a fall/winter technique.
Seasoning food appropriately is so important. Michael Ruhlman's new book "Twenty" really gets into how and when to salt, so read that if you can.