If you are not a natural extrovert (I'm not), I would say, at events:
Get there early, when there are few people there. They will all be hanging about and it's easier to start chatting. Or talk to the organiser - how many people are you expecting, I've been looking forward to hearing X speak for a long time - just social chitchat.
Find someone on the periphery who looks as awkward and nervous as you feel. Ask them something about why they're here, what they do etc etc.
If someone you want to speak to is going to be there, and you don't know them, ask the organiser if he/she knows them and can point them out.
Women are often more approachable and friendly than men.
The only time I have failed in this was when I was invited, some years back, to an Arab country's National Day reception. I was the only woman there, and absolutely no-one wanted to speak to me. I had an awkward conversation with an academic who was hiding behind a pillar, gave up and went back to the office.
For work contacts:
Pick up the phone and ask someone's advice - X says you are the expert/go to person on this subject, and I hope you don't mind if I pick your brains. Most people will be flattered by this. You can try email as well, but a phone call is often more fruitful. If they are very senior, they (or their PA) may suggest you speak to someone else on their team instead. Thank them, and do so - this way you'll have an "in" to what they do anyway.
Invite someone out to lunch, if you have an entertainment budget, or for coffee. I managed to get to see a woman who was refusing to see me unless I came to her office (for various, political reasons, I wasn't allowed to go to her office; for various political reasons, it was in her interest to get me there) by calling her from the cafe across the street and asking her to meet me. She agreed, and we got a great professional relationship going.
LinkedIn:
Can be useful if you join work related groups and enter into discussions on what you're working on, identifying someone who could be useful/someone you might want to work with.