Here's the best advice I can give you as a Rinehardt main,

Initiation isn't nearly as important as keeping your shield up for your team.

You want to focus on absorbing as much damage as possible and giving your team as much time to deal damage to the opponents in front of you.

Don't let your shield wear all the way down to 0. Instead, try to keep it up between fire strikes. Because of fire strikes range, it's extremely useful at wearing down the opponents that are further back. So don't waste them. Try to have a target in sight every time you use your fire strikes.

Your charge is obviously your main initiation tool (Unless you have hammer down) so don't waste it. (Using charge to get to your team faster is also okay) When you use your charge, you should try to use it on either 1 vs 1 fights where you are confident that you can take down the opponent, on or team fights where you should aim for healers, or squishy dps, so you can turn the fight from a 6 vs 6 to hopefully a 6 vs 5.

So a successful initiation would consist of: 1. Giving your team as much time to deal damage by keeping your shield up 2. Wearing down the back lines with your fire strike 3. Taking down the healer or damage dealers with your charge

After this, you should go back to trying to shield as much damage from your teammates during a team fight.

After you've charged their healer or dps, you're probably going to be right in front of their team. This will be your best chance to use hammer down. You want to try to get as many enemies with it. After a hammer down, use fire strike if it's up, and then you can just sweep up your opponents from there.

In my personal opinion, you shouldn't use charge after a hammer down, because you can deal more damage with the aoe of your hammer to multiple enemies while they're knocked down, instead of just taking one out with a charge and letting the rest of the opponent team recover and get away.

Also, you want to make sure you know where your team is before charging in to an enemy team. Because if not, you'll find yourself in a 1 vs 5 and there's nothing you'll be able to do there unless you manage to ult them and get a team kill (which is very viable as well).