Packs: I've had great luck with Osprey packs--they have the best warranty in the business (lifetime, transferable, no receipt necessary, no questions asked), are super durable, comfortable, and reasonably priced. The Kestrel is my go-to day pack.
Compass: +1 for Suunto
Fire Starter: I carry at least two methods. A Bic lighter and a ferro rod are my go-to (Nanostriker XL, but just about any will work). For tinder, cotton balls soaked in Vaseline stored in a pill bottle are about as good as it gets.
Jacket: I carry a Marmot Super Mica because it's lightweight, but some good brands are: Arc'teryx, Patagonia, Marmot, Mountain Hardware, and Columbia. Just be warned that anything waterproof will need to have zippered vents or you will get soaked with sweat while hiking. Even Goretex will leave you wet and clammy if you are really hiking hard. The only semi-exceptions are eVent and DryQElite, but they are expensive and still not as breathable as vents.
Other items: Headlamps are super convenient and if you are doing full-day hikes, sooner or later you will find yourself hiking in the dark. I also always carry a mini first aid kit, or at least a blister kit (mole foam, athletic tape, hand sanitizer). An extra pair of socks is nice if you get your other ones wet, as wet feet usually lead to blisters. A good pair of well-fitting boots or trail runners is probably your most important piece of gear, and will be the hardest to find. The key is to find a very knowledgeable salesperson to help out--and they can be even harder to find. Short of that, search online for how to choose hiking boots--it's a topic that takes a few pages to really cover. Blisters are more of a problem for backpackers, but they can definitely happen on day hikes too, and they can be seriously debilitating.