I'm sort of in this position at the moment as I have just moved country and as I have a small child I feel I need a support network now - I don't want to wait and form friendships organically.
It is hard work. I have done it before. You can't just sit and wait for friends to magically accumulate. You have to almost "date" people, make an effort, seek out contact and then keep it going. It can be exhausting, it can be the last thing you want to do on a particular day but if you want to make the friends then you need to keep this up, for probably I would say 6 months or so before you start feeling the rewards from it, and then it takes about 2 years in my experience before you can call somebody a close friend.
So, pick a group, any group - in my opinion, the bigger the better because it increases the "pool" so you're more likely to find someone you get along with, although small ones can be better at forcing you to actually talk to the others rather than hiding away in a corner. Someone mentioned meetup.com which is actually very good - I found a good english-speaking meet up group on my local one, obviously this is pretty restricted to being abroad, but you may find something! You really have to decide if your motivation to join a group is for something to do for yourself, or to meet people. If you are just doing it to meet people, then a passing interest in the subject is enough.
Then, every time you speak to someone, try to remember their name, one thing about how they look and one piece of information they tell you, especially something about an ongoing situation is good. This means that next time you see them, you're more likely to be able to put the name to the face, and you can ask them "How's the new job going?" or whatever it was they told you.
After a few weeks, you will start to feel like you "know" some of them better than the others. This might be a good time to suggest meeting outside of the group, or ask/offer specific knowledge, like asking their recommendations for a restaurant/how to cook a particular food, offering to lend them a book or DVD, that kind of thing. Offer/ask to add them on facebook as soon as this seems relevant.
Then you just have to make the effort to go along to everything possible even if it seems like more than you would usually do, just for the opportunity to talk more with people and build relationships with you. Beware of latching too much onto the first person who is friendly to you as well. Often they are not the same person that you eventually end up being friends with, so keep your options open!
Once you've met up with people once or twice start inviting them out yourself, either out or over to yours. This can feel forced and awkward at first, remember, they aren't old friends you've known for years, you are "dating" them. You should always be yourself rather than trying to impress them or putting on any kind of false persona, but push yourself to do more meet ups etc than you usually would. It pays off :)