You might try the following dialog: "Hi (insert manager's name here), do you have a free moment? (then if they do...) I've really enjoyed working here so far, and I just really wanted to thank you for the chance. I was also wondering if there was any possibility of increasing my hours. I understand if there isn't, but in case there is, I thought I'd ask."
It helps to also offer to work extra shifts that are not popular with extra employees, or offer to fill in for people who are sick or on vacation. Also, during your regular shift, make sure you are ALWAYS working--even if it's slow and there isn't anything you have to be doing at the moment. If your manager seems busy (say he or she is unloading some boxes), ask if they would like any assistance; same goes for your fellow coworkers. If your manager sees you as a go-getter that they can depend on, they'll want you there as much as possible.
Here's another sample dialog you could try:
"Hi (insert name here). Do you have a free second? I just wanted to thank you for hiring me. I've really enjoyed working here so far and was wondering if there was anything I could do to improve my performance. (then wait for their response and if it seems appropriate to continue...) I also wanted to check in with you to see if there was any possibility of increasing my hours, now that I've been trained. I don't mind working (insert least desirable shift here, aka nights and weekends) or filling in for people if they call in sick. If not, I totally understand."
Avoid going into how you need more money to pay your bills--you want the employer to feel like you'd be helping them out, not the other way around. It helps to keep this in the back of your mind: "How can I as the employee make your life easier?" As long as you're respectful and avoid interrupting them when they're busy and stressed (unless, of course, it's to offer to help), you shouldn't seem out of line at all.
Also if I can't get more hours, I will have to work 2 jobs at the same time. would it be appropriate for me to tell my boss I will be job hunting for an additional part time job elsewhere? Basically, What should I do if I'm not given more hours?
I think i don't have to be afraid to ask for more hours, Tell them plan and simple that you need a bigger paycheck to pay your bills and if they can't provide it for you, explain to them that you will search for another part time job.
It's probably not best to tell your boss point blank that you'll have to look for an additional pt job elsewhere, at this point. You could explain that you need to be working (ex: 30) hours per week to pay your bills/for school, and that you'd love to work all those hours here, if it's possible to get more hours. Be very matter-of-fact about it, without going into it too much. Employers can respect that you need to pay your bills, but they don't want to hear a lengthy tale about how your school has raised tuition or your car keeps breaking down.
If your manager says "sure, we could use some extra help on these nights," then great. If he/she says "sorry, I can't give you more hours since that will just take away from others who also need the hours," then be gracious and thank him/her, and then ask which hours would he/she prefer you keep open for working at the restaurant. If your schedule has been pretty consistent, you can ask if that will continue to be your schedule. But some restaurant managers like to change up people's schedules with no warning--although rude to employees, it's important to make sure whether getting a second job will conflict with the one you have.
I am a new member and this post is a bit older, were you able to get more hours at work? I hope so! You got a lot of great tips. It seems really hard to outright ask for more hours, but managers (and this is coming from a former retail store manager) love to have employees on the clock who actually want to work more.