There is a professor who is willing to allot an open position in the lab but he's just delaying about it. He just won't tell me what project is he going to assign to me. He told me he wants to talk to other people and see if their projects are more relevant to me. I want to politely ask him to speed up the process. How do I do that?
Professors are notorious for lacking a sense of urgency in these types of areas of: logistics, operations, bureaucracy, red tape, paperwork, etc. Rarely do they understand things from a business perspective. Some are better than others, but they're not common. Also, believe it or not, they are very busy people. They do more than just teach a few classes.
Take these two factors into consideration and realize that you need to prompt them more frequently. Just ask if they've been able to look into your request. If they say, "I'll look into it next week." don't wait for next week to finish. Ask in the middle of the week. Again, they need prompting and if you're not doing it, they'll just attend to some other task that they may feel is more pressing.
At some point, the professor could get irritated by your continuous prompting, but you really have no choice if you want to get this assignment completed. You can always back off on the level of prompting. Do not say things like, "You said you would do it last week." or try to hold them to some other verbal commitment. You're not their boss and you're just one of a dozen requests they are trying to manage. Just indicate you want to remind them. Keep it simple, nice, non-threatening or sounding like you're entitled to any kind of quick action.