Hello. I work at the University of Florida and thought I could provide some insight.
The only home Gator game this year in October is October 1st against Alabama. This game will also probably be the most-attended home game this season and tickets will be harder to come by. Even in "down" years, for SEC Conference games the stadium will likely sell out. You will have most success purchasing your tickets through a re-sale broker rather than from the box office directly. You could come to Gainesville and purchase tickets from a person on the day of the game but there are no guarantees that you will get seats together or a good price. Make sure you do not purchase student tickets as you will not be allowed in without a student ID. If you do come to Gainesville, come to town several hours before kickoff. Parking near the stadium is extremely limited and you will likely end up parking far away and walking.
You could also go to the Florida-Georgia game in Jacksonville on the last Saturday in October. Although it is not quite the same as seeing a game in the Swamp, this is one of the classic SEC rivalries.
Going to a college football game is fun, but it is HOT and LOUD. Most games are played in the middle of the afternoon and even in October it can be uncomfortably hot. You can't bring in any backpacks, food items, or anything other than a bottle of water.
There are some great traditions associated with college football, and it would be an experience for you to say the least!