There are three types of climbing: top rope (where your rope is already secured at the top of your climb), lead (where you place protection and rope as you climb), and bouldering (no rope because you're climbing between 10 to 40 feet). And if you get really specific, there are different types of climbing within each of these (free climbing, trad, sport, etc). Climbing is rated by different scales. Although all are different, the higher the rating, the more challenging the climb. And challenging climbs get easier with perfected techniques. And techniques can only be perfected with practice. This is why climbing can get addictive. Invest in the right gear as you advance.
Something comfortable, ideally stretchy and/or durable, that can withstand abrasive rock. Keep in mind you will be in a harness the entire time, so comfort is emphasized.
Two pairs: one is called your approach shoe. It can be as simple as flip flops or as fancy as grippy-soled hikers. It depends on where you're going.
Your second pair are your climbing shoes. Climbing shoes are very specific to the type of climbing you'll be doing, but anyone who is not advanced will do fine with all-purpose climbers. You can often rent these from your guide.
Climbing accessories can be expensive and unless you're advanced enough to go climbing in the dark, it's better to rely on your guide to provide you everything you need. Recommend bringing items you typically use based on the season you're climbing in (see above). However, accessories like harnesses, chalk and chalk bag, and of course, rope are typically provided by your guide.
SAME AS CLIMBING
Ice climbing is not as dangerous as it looks. All your gear can be rented from your guide, but here's a pro-tip: schedule your trips during "ice climbing festivals" where you can try all kinds of new gear. There are, of course, different types of ice climbing, (like alpining, mixed, etc) but you shouldn't worry about that unless you plan to go pro.
Sturdy sneakers or hiking boots.
You would also be outfitted with specialized ice climbing boots by our guide if you don't already have your own.
Additionally, you will also be outfitted with specialized crampons that go over your boot.
Winter accessories (see above)
You will also be outfitted with harness, helmet, and ice tools (like ice picks, but awesomer) by your guide, unless you already have your own.