They all start and finish at almost the same time. theres no post season - only being #1, although each league has cups (or tournaments) also along with the regular season, along with the Champions League where only the top teams in Europe compete
As a follow-up, leagues work in that whoever has the most points at the very end of it all wins the title, there are no playoffs. Tournaments play separately as their own titles. When you hear about someone getting a 'Treble', they are winning three significant trophies. Every professional league has its own tournaments that work in playoff style, some more than others.
There are also a lot of european-wide tournaments. Certain leagues are given a number of spots depending on popularity and strength. The highest level is the Champion's league, which is the highest of every league, sometimes like the top 4 from certain leagues. They go in group stages, like in this World Cup. Next is the Europa league, which is like the next best in the league. A lot of times teams will be playing for these spots to try and prove themselves.
Skills shown in league games by players can get them called up to their respective international teams, where they normally play in qualifying groups that are drawn, or friendlies. World Cup and Euro Cup are the biggest in Europe (duh). Different nationalities play on different teams, like in the states, based on the trading system. There are deadlines that are universal, at the start of the season in late summer is the first transfer period. The second starts around January.
As I said before, I can answer any questions you have if you PM me. I don't want to crowd up this thread any more than needed.