Welcome! I started at UH last August, and I'm having a great time so far.
If you run, I know some runners among the faculty and can put you in touch with them. You can DM me if you want to know more.
For cycling, there's Honolulu Bicycling League:
Hawaii Bicycling League
I don't know how active they are, but they organize a century ride every year (I think). I hear they're also working to get Honolulu more bike-friendly, which is why I've been thinking about getting involved. I'm not a hard-core cyclist, but I would love to ride my bike to campus without feeling like I'm endangering my life (riding in the street) or the lives of others (if I choose the sidewalk).
I hike a lot, and I see mountain bikers all the time on the trails. I don't know the details. Your best bet would be to pop into one of the bike stores when you get here and ask around. There's a nonprofit bike refurb place on campus called Cycle Manoa, so that might be a good place to ask.
Cycle Manoa Home — UHM Sustainability (http://sustainability.hawaii.edu/group/cyclemanoa - broken link)
I've used McCully Bicycle for assembling my bike when I got here and some small repairs. They've been great, and would probably have info about local groups.
McCully Bicycle and Sporting Goods - Welcome!!!
As for horses, I know there are stables Waimanalo. I was driving a visitor around the other day, and we saw some folks riding along the road. I don't know anything about them, but if you google "Waimanalo stables" you get about four hits. You can get their numbers and make some calls.
Best of luck, and e komo mai!