I agree with Fishwife, I have seen the trauma caused when children get attached to a series male figures who turn out to only be fleeting parts of their lives. It happened to friends of mine in childhood and I'd say the effects have lasted throughout the way they deal with adult relationships today.
I was a single mum for 13 years before I got married to DH, and waited a year before I introduced him to the dc. I would say that you have to be very careful about their commitment before moving in, especially if your financial situation means you'd lose benefits or tax credits. A partner is expected to support you and any child in the household regardless of biological relationship, and not all men will be able to do that financially or be willing to. I think it's a very demeaning policy for women, as they can often be left completely dependent on a man, but that's the way it is.
You have to be very careful when it comes to moving in, if you're moving into a man's owned house, because that gives you and your children no rights at all if there is a relationship breakdown. I made it clear to DH that I would not move in with him until we married, because married women have rights to stay in the house in the event of a relationship breakdown, whereas a co-habitee would not.
I met DH through a sport, I think hobbies and activities are really good for meeting people, as you tend to get introduced in a group, which means you can see a man interact in all sorts of situations. I tried internet dating too but although I had lots of fun with it, I found it less useful for meeting suitable long-term partners. It's absolutely fine though, to meet men for sex/fun and not see all of them as potential father figures for your children! But if you're going to do that, definitely don't introduce them at any point.