Making the first move in any type of relationship can be difficult regardless of gender. In some parts of our society, it is considered poor taste or a forward action for the woman or girl to make the initial move in a romantic relationship. However, if the one she is interested in is a bit on the shy side, the move may never happen if she doesn't go first.
The reality is that most guys like the ego boost from thinking that a girl is interested in them. Unfortunately, most guys don't take subtle hints too well. This leaves it up to the girl to do something obvious if she wants to see where things might head. So, the dilemma is to make the first move without coming across as desperate or easy.
The first move doesn't always have to be physical or verbal contact. It can be as simple as making sure that you have several times where you end up in the same place at the same time. A smile or nod may be all that happens at first. By the third or fourth meeting, he should be figuring that you two have sort of met. This will probably give him the confidence to speak first. Once the ice is broken, the limits are raised.
At this point, the girl can carry the conversation into somewhat personal territory. As the two of you get better acquainted, there is no problem with opening a door or two about what you like to do on a date. This gives him some ideas of what type of date to invite you on. If after about 3 or 4 conversational visits he hasn't moved, you can do the invite. Go for something simple like getting a cup of coffee tomorrow morning at a nice place. This is also inexpensive in case he still doesn't smell date and expects to split the check.
If you have a couple of these "friendship" outings, grab a handful of money and ask him out on a real date. If he balks, you may have to offer to pay. It may be that he hasn't asked because his finances are limited. If it appears that he's just cheap, move on down the road.
Many girls find it is better to make the first move through a friend. That way if he isn't interested, your friend can buffer the bad news on the return trip. It also puts him in the position to actually ask you out with the confidence that you will say yes. He may have had some recent rejection issues and be a little gun shy.
If none of this appeals to you, ask him for his email or instant messenger ID. Then, pursue it that way. You can often say things online that just don't seem to work in person. It may also give him the bump he needs to make a move.
If you are aggressive and liberated, just walk up to him and ask him out. Most men will say yes much more easily than most women. In fact, most men dream of this scenario. You can work out how the tab is handled later on.
The main thing is to make the move in whatever way you are most comfortable. Doing nothing when you are interested is not a good option.
You may want to inquire to people who know him better before making the move. This will help keep you from wasting your time on someone who has traits that you find unacceptable. It also will help you know his likes and dislikes. That way you won't be as likely to say something offensive or stupid on the first date.