I started kindergarten at 4 (late birthday = October) because I had been reading for nearly two years. My parents made efforts to have me skip 5th, but the school was very much against it (used my late birthday as one reason) and so provided transportation to another school 1 or 2x/week for a fun enrichment class. Though I enjoyed the class quite a lot, I really wish I had been skipped.
In early elementary school, I enjoyed being "number 1" in all subjects in the classroom. Unfortunately, it didn't take me long to discover that I didn't need to put in much effort to maintain my status and still be No1.

This lack of effort and attitude hurt me in high school and especially in college as I came to recognize that I didn't know how to study. Everything had always come so naturally in the past.
Our DS13 was offered a mid-year skip from 1st to 2nd. The teacher situation would have been awful, so we didn't do it. So, the next school year they were going to move him to 3rd instead of 2nd, and we (regretfully) decided not to do it for a number of reasons: others sharing horror stories about skipping, misguided info on skipping gifted kids, but mostly because we decided it wouldn't be a long-term solution. We were afraid that it would have only been a temporary fix and we'd end up needing to skip him again. So, we tested him for two competive private schools. They both wanted him. One said they'd start him in 3rd, the other started all their students a year late anyway, so we put him there in their 2nd grade class. Unfortunately, it was a very poor match. I ended up homeschooling him after the winter break.
Hindsight - we wish we would have kept him in the very good public school and skipped him into 3rd. When in 7th, his counselor recommended a skip to 9th, and the principal concurred. Unfortunately, the district rejected it completely and commented that that's why they have honors classes. It is frustrating.
We have learned to take each year as we go. Nothing is set in stone, and his input is very important. If he hadn't gotten "double-bumped" (two years ahead) in math, he had asked to be home-schooled or virtual schooled (and we would have started him in 9th.)
I learned from Dr. Rick Olenchek to listen and discuss with your kids the ramifications of their educational pursuits. What do they really want? You have to sift through some of the things they say, and you also have to understand them and their strengths and weaknesses.
I wish you the best of luck!! Sorry for the long post!