You can compost both grounds and filter. Used grounds are fine around plants as a mulch, they do not have much acid since it ends up in the coffee cup, but they certainly won't hurt plants and many people love them for roses and all kinds of plants.

I use a 1-gal ice cream bucket with a lid. Print out a nice sign that says "Used Coffee Grounds, Filters and Tea Bags for Composting." Have two of them so you can take one home every day or two. You can get a lot from an office coffee machine!

HOOK! Reel her gently.....gently....gently,
Oh! madaktude! I didn't know you were here!
You'll have fun reading through these forums and learning all about composting! Especially during dark cold NY nights!
Your coffee grounds, kitchen waste, paper towels, neighbors' leaves and more will become your friends! I'm sure by spring you'll be hooked (line and sinker!) Nancy

haha. If I had a nickel for every exclamation mark in that post, I could buy madakatude a ticket to get off the composting crazy train before it's too late. :-D

Seriously though, start small, do only what you can handle, and if you get too many coffee grounds, hide the bucket for a couple of days. Also, buckets of coffee grounds do not get icky and smelly that fast like other kitchen scrap can, so it's a very nice and convenient composting ingredient to work with.