Yes, there is no break, new season starts right away and your players will have one star less. That means they are less competitive on higher level, not that they are losing quality (as some might wrong understand).
If you have expensive 3 or 4 stars players which aren't useful for you, sell them. After season end, almost every player loses on value (expect oldest players) and gets 1 year older. Also next season starts Champions League. There will be groups like in real CL, 4 teams in group, 2 of them will go into knock-out phase.
Don't do mad training. Besides that, you don't need such powerful team for low levels such lvl1. You could have won level 1 league even with 4* team. There are few useful threads and posts about game, training, players, formations etc. on this forum, (unless some authors removed content) google-search them.
Next season I suggest you to buy 5* players which have sale price under 600K. That way you can get a lot of excellent players for reasonable money. Always check form of your players, if form is bad (6 or less) then try to change their position and test in friendly, if form doesn't change, sell that player no matter how high quallity he has.
Good Luck and have fun!