I have been biking for about a month and really enjoy it. However my legs just give out on me. I run 6 miles two days a week then bike 6-8 3 to 4 days a week. Any suggestions on how I can improve would be great. Oh, not sure if this matters but I'm not a big eater in the mornings before I go biking. I eat a protein bar and have water to drink.
It sounds like your legs might need some rest. While running and biking make use of different muscles in the leg, there is still some overlap between both activities.
My suggestion would be to take at least two full rest days before you go biking again. Then after that ride take another day of rest before you do another activity (running or biking). Try to focus on rest days: Four days of rest, three days of activity.
Before your ride you should consume carbohydrates (a mixture of simple and complex carbs) instead of protein to supply your muscles with energy. I personally would prefer something normal to eat, eg bread or pasta, as energy bars are quite expensive in the long term.
You've only been biking for about a month; it takes time to build up cycling-specific muscles.
How fast you spin your legs is known as "cadence". You should aim for a cadence of approximately 90 RPM; you can get cyclecomputers that measure it directly or estimate it using a watch. Novices generally ride at a cadence that is too slow; that requires much more strength and can easily burn your legs out. Shift down so that you ride at a reasonable cadence and don't have to push very hard.