Car is not starting randomly.

Yesterday when I left work my car was hard to start. It would not turn over
right away. Got it going and went to pick up my son. Turned my car off and
it started right away. I filled up at a gas station and it would not start up
again. Nothing not a sound. They noticed my battery is severely corroded.
They got it off and cleaned it up. Still nothing. I got it jumped and it started
right up. I left and had my dad meet me, we turned the car off and it started
right up again. The evening I went to the store and it was fine. This
morning however, it was hard to start again took a couple turns and my
clock was wrong (I reset it after the up I got s it was right that night). Got to
work just fine (about a 30 minute drive), went out on break to test and see
if it would start and nothing. I am going to get a jump. I am told it is
probably the battery. Should getting a new battery fix this problem?
Someone thought alternator, but I am told that if that was going out my
radio would not work or my lights would dim...nothing like that is happening.