Agreeing with others, a good comfortable backpack is very important (mine is an osprey, very similar to this model, lightweight, and has lasted through seven years of heavy use), as are good sturdy boots, preferably with ankle support. You can get a pack online (features that I find useful: external water bottle pockets, at least one main zippered pocket and one small one, and external bungees so if you take off an outer layer/raincoat you can stow it somewhere where it'll be quickly accessible) but I'd definitely recommend going into a store to try on boots, if at all possible. I have Merrell boots that are mid-height (the really tall ones aren't necessary for most people and most terrains), and I've found them very reliable. Check out REI's day hiking section for some options.
As for what's in my bag, for a day hike (2-8 hrs) I pack: a mini first aid kit (just a Ziploc containing band-aids, alcohol swabs, ace bandage, and little baggie of ibuprofen), lightweight emergency blanket and emergency whistle (with waterproof compartment for matches, striking surface, and mirror, like this one, although I would not rely on that cheap compass), water bottle (at least 32 oz, mine is a Nalgene, but it honestly doesn't matter as long as it holds water), hat (baseball cap for sun/bugs or wool hat for rain/cold), wet wipes/purell and baggie with a small amount of toilet paper (if you use it, make sure to dig a small hole well away from water and from the trail), lightweight flashlight/headlamp, knife (mid-size swiss army or similar, kept sharp and clean), food (either just a granola bar for a short hike, or a lunch of bread/apple/hard cheese/sausage/chocolate, all of which pack well), and, if rain is forecast, a raincoat and baggie for my phone (they also sell rain covers for backpacks, and I highly recommend getting one if you hike frequently in rainy areas).
If you're doing steep inclines, or rough terrain, you might want to also pick up some trekking poles; although not everyone uses them, they can reduce strain on knees on the downhills, and help improve balance on loose/shifting rocks and thus reduce injury (I've been hiking/camping all my life and only just added some to my gear, so they definitely aren't essential unless you have a preexisting knee injury, but they're certainly nice to have).
As far as camping gear goes, what you get depends VERY much on whether you're primarily car camping or primarily backpacking. If you're planning on doing multi-day hikes, or hiking in very remote areas, you might also want to add some navigation gear (compass, paper maps, GPS), but they're not essential if you're on established trails in reasonably well-used areas.