BumWad - there's not much to get your head around.
Basically, it's very similar to how it was. One team holds a gym, you can battle that gym with your Pokémon, and instead of the prestige counter going down, the enemy Pokémon's happiness rating goes down (as indicated by a heart graphic above each Pokémon). Once you have battled the gym down, you can put your Pokémon in the gym.
If your (or other team member's) Pokémon are not completely happy, their heart will be a little bit empty, so you can feed them a berry. The less happy a Pokémon is, the less well it can fight, so it makes sense for the team to feed them.
Coins are now awarded on an hourly basis, and you get them when a Pokémon comes home to you after being defeated. You get one coin per hour, apparently.
You can also spin the gyms as pokestops. First, click into the gym, then you will see a little photo disc icon. Click this and you can then spin the photo disc.
Raids are not really in effect right now, so don't worry about that yet. Apart from that, nothing has changed. You still battle in the same way, is clicking the hell out of the opposing Pokémon and doing an occasional special attack.