I've just been replaying Saints Row: Gat Out of Hell today, and this time I noticed something odd about the early missions (in which you recruit your allies and gain your powers): They're in the wrong order. Now you can actually do them in any order you like, but the game prompts you to do them a certain way, specifically Blackbeard/summon, Shakespeare/stomp, Vlad the Impaler/aura and then the DeWynters/blast.
There are several reasons this is dumb. The first is that summon is actually pretty worthless, so having it first won't really help you in any of the fights that follow, especially without any additional elements or upgrades. Stomp is the most useful power and the one I went for first, but that's not actually the best way to go either.
My advice is to start with the DeWynters. Their mission is probably the easiest, since you have two homies to help you and it's just a straight-up fight against a few handfuls of demons, with plenty of room to run away and hide if you need to. I died once to the legionnaire that comes in at the end, but only because I forgot to sprint/fly away when he charged me.
After that you'll have the stone blast power, which will be useful for the remaining missions. The next I'd recommend doing is Vlad's. Aura is another basically worthless power, but his mission is super easy once you've picked up some guns from the previous one and the blast power. There are some shadow demons who can be a pain, but they can be turned to stone to make things easier; a marked improvement over the "official" order, as summon and stomp are both useless against them.
Next up is Shakespeare, and the hardest of the lot. More shadow demons, and in an enclosed space where it's hard to get a shot off without putting yourself in the line of fire or coming too close and making them teleport away. You can kind of cheese this bit by hiding behind the speakers and peeking out just enough that you can see them but they can't see you, but it's a pain. Again, stone blast is going to be invaluable. This fight also includes a dark inciter, who is a massive pain in the arse if you don't have blast yet.
And last is Blackbeard. This fight is actually really easy - but only if you already have the stomp power. It was by far the least fun one to do my first time through, because imps are small and quick and hard to hit. They're not likely to kill you, but they're really annoying. About half way through you get the gods hammer, which is also great at dealing with imps, but it really should have been given to you at the start of the mission.
The next thing to do, obviously, is to go find the altar that gives you the vacuum element for your stomp power, because you're basically going to want to be using that almost all the time for the rest of the game.
For more on Saints Row: Gat Out of Hell, see my comprehensive review.