Mid game is the starting point of real teamwork and crucial for getting objectives. When should you group, and when is the best time to start team fights?
Mid game start around 15:00 minutes to 25:00 but you can start earlier if the tier 1 tower has been destroyed, during this time you should be grouping with your team. In some cases the team that groups earlier than the other one tend to win in objective and in kills, because this is a team game not a free for all. You also need to keep in mind that if you group, that the other team can easily "back door" and that you may lose experiences/gold in other lanes, because you get less experiences for everyone that is in the lane when you or someone else kills a minion.
What i mean by "back door" is when one or more opponents goes in other lanes to kill the towers, this can be very effective if the champion is good at killing towers and can fight if needed or easily get out of getting "caught". Some teams have better team comp then others making it more difficult to win team fights, sometimes splitting them apart is the best way of winning fights. Champions with "AOE" abilities are normally stronger in teamfights then a champion with only single target abilities, "AOE"=(Area Of Effect) means it can target more than one champion.
When you start grouping the goal is to get as many objective as possible, objective can be kills, towers, "inhibitors", "dragon" and "baron"/"Herald". Only start a team fight if the odds are in your favour, by this i mean only fighting when you outnumber them or if you catch out one of their "carries". By "carry" i mean the opponent champion that does the most damage on the team, normally it is the "adc" or "mid laner" and they are usually the weakest/"squishy". If you can catch someone out of place before a team fight occurs you will have a much better chance of winning the team fight, as long as you do it before the opponent champion can respond.
Most of the mid game is spent "seeging", what "seeging" means is when you are at your opponent's tower or your tower and you can't start a team fight because they may be too strong for you to fight under a tower or may be too strong for you to fight normally. Some champions are great for "seeging" because they can clear minion waves fast or "poke" the enemy champions from a distance, champions with "AOE" abilities and rages are the best for "seeging". If you find yourself "seeging" make sure you play it according to your teams comp.
Note: Some champions should not "engage" team fights and some are great at "engaging" team fights make sure you know what your champion is build to do. Normally tanks with high "cc" are the best at starting team fights and the champion that does damage should not start team fights."cc" means (Crowd Control) a ability/passive that can or will (stun, taunt, knock up, snare, slow, silence, fear, pull)