Starter problem?

If the weather around here gets below freezing overnight (and it does almost all the time) my car has a real hard time starting. I have to turn the key and hold it there for longer than one or two minutes because I get absolutely no clicking noise or anything until I hold it there forever. I get the radio playing and the heater going, but when I turn it, there's no noise. Sometimes I leave the key in the 'on' position for about 5 to 10 minutes and I'll finally get something. Is there a problem with my starter, or my starter relay, or something else?

Also, when it does finally start, I have to give it a lot of gas until it's running like normal. The engine shakes really bad at first and it's almost like all my cylinders aren't firing. I don't know if it's a compression problem or what, but it doesn't like to start after it gets cold. If I don't give it gas for about 10 seconds, it'll stall and I have to try starting it again. If you need more info on this problem, just let me know and I'll give you more info. Thanks everyone.