At the start every run will look similar as the heroes will be the same each time. This game isn't very complicated so the mechanics don't really require explanation.

As such don't sweat the start of the game too much your early goals will be to get your first character to level 5 and craft a full set of gear for him and a 2nd companion. The 2nd character should allow you to clear the level and move onto T2.

Don't worry about competitions yet, they're honestly just a waste of time. You can also safely ignore all the green side quests - their rewards is barely worth it. In the middle of the T2 area or start of T3 area you get a quest that will increase your competition reputation bar to nearly max when you finish it, use this to save a bunch of time in the early mid game by skipping those worthless competitions!

The way to speed up T1 and T2 into the later stages is to focus on crafting and selling gear. The higher the quality the better so remember to rest at the spa when it drops by 1 or 2 levels as that will impact crafting quality. Craft magic gear, smithy gear and hunting shack gear. Then get back at the grind. The important item in this area is in the Magic shop, get a Masters Cap with high Vampirism for everyone to keep it moving.

In the T3 area the most important item to focus on are space armor with maximum gold steal and the Massive Shotgun also with gold steal. I don't run the wolf helm for the added of 50 max gold per kill b/c of the regen on the cap. The idea here is that you're getting as much extra gold as possible while also not stopping or bleeding money for heals.

Grind grind grind. Buy levels, stats etc. This should move you into the T3 area which is a good difficulty jump but also an important area to grind. At this point you should have the 3 teams running, 1 person solo clearing the 1st level with as much gold steal as possible. Team two should be a team of two clearing T2 and your strongest team clearing the new T3 areas when possible.

Note: Keep your stamina high and damage output low, the more times you hit the enemy the more gold you steal from them.

Doing this grind the game should fly by. It is slow by design but this should help you speed things up considerably.