Intermittent starting issue

Okay, the last week three cells in the battery went out so I went and bought a brand new one put it in and everything was fine for a day. Then when I went to start it it would not start right away, tried three times before it started and the rpm's jumped up to 4,500 then came back down. It idled for a few seconds then the engine shut off, but I never lost power (electrical). I tried to start it again and it would not start just kept turning over and over but no start. So after trying three times I let it sit fifteen minutes and tried it again and it started and shut off. The same I waited fifteen minutes. A friend told me next time I try it to hit the gas a little and so I did and it fired right up. So I stayed on the gas a little and it did not shut off, but when I let of the gas it shut off after a few seconds of idle. So I tried again and it fired up again when I was hitting the gas a little so I let the engine warm up and then let off the gas and it stayed idling for ten to fifteen minutes then shut off again and would not fire back up right away. So I waited fifteen minutes and tried again and it fired up.
So I took it up the road and back and it did fine. Started up fine afterwards but when I went out today it would not start no matter what I tried. Please someone help me.