Starcraft – the meaning of the number 54 in Blizzard’s games

You start out in the practice league. In this league you play unranked matches against other practice league players, on special maps designed to prevent early rushes. Matches in the practice league also take place at a slower game speed.

You can play 50 games in the practice league before you have to move on. You can also choose to leave the practice league early, but if you do so you won't be able to go back. (This is the "Skip" button you mentioned)

Once you leave the practice league, you begin a series of 1v1 placement matches to determine which 1v1 league you will be placed in (Bronze, Silver, Gold, etc.) You can't play any ranked games (1v1, 2v2, etc) until you leave the practice league.

If you scroll down to the bottom you will see a button that says "Show more entries". Keep clicking this to see all of the games to my understanding. It may stop eventually, however.