Just say, "hey, do you wanna do _____?"
I used to be shy (and now I can't shut the f*#k up), and one thing I learned is that when you never initiate anything, people stop asking you to do things because they assume you are not interested.
Here are a few things I learned about how not to be shy:
1. If you have nothing to say, Ask Questions!! This is the #1 rule.
2. There is nothing wrong with silence! Sometimes there is just nothing to say, and that is perfectly ok. In other cultures, silence is a perfectly normal part of communication. It's only in America that we feel like we have to blather on endlessly and not have a moment of peace.
3. Get out of your own head, and actually be interested in other people. When you stop thinking about yourself it actually becomes quite easy not to be shy. "What are they thinking about ME?" "What do I say?" "Do they like ME?" etc. etc. etc. -- I, I, I, me, me, me -- that's the monologue going on in a shy person's head.
You will not click or be great friends with everybody. Allow yourself to be boring. You are not obligated to be interesting or witty or entertain people.
There. I just gave you permission to be the most boring person on earth. Now go out there and see what other people are all about, and if it occurs to you to invite them to do something, then do it. If they can, they probably will. If they can't then do it another time.
The best thing that ever happened to me is that I stopped being shy and insecure. There's a big world out there. Embrace it.