I'd like to thank TheAngelofDeath for finding what I'm convinced is the best strategy for getting to the center of the galaxy and the best path to becoming, essentially, unstoppable. Thanks TheAngelofDeath!
First I must confess I wanted to test this strategy of allying with the Grox as quick as possible so I used the money and health/energy cheats. But I'm sure it can be done without cheats as well and I'll be starting a non-cheat game soon now that I know this great strategy.
In my game, before I went looking for the Grox I got to Omnipotent and made allies with several civs of which at least 4 where lvl 5. Then I started hopping wormholes.
First time I found the Grox I was surrounded and there wasn't any other civ to use the planet busters on so I went back and hopped some more wormholes. Eventually I found a wormhole that ported me close to the Grox with other civs close by. So I used the planet busters on a couple of those poor unfortunate civs. Funny how sometimes after I released a nuke they would contact me and try to surrender. Sorry mwahaha!
I believe you have to planet bust at least 5 planets to get the +50 with the Grox however I did try nuking several uninhabited planets too. That may be all that's necessary if you don't want to kill innocent civs, but I'm not sure. I will test that on my next game if someone else doesn't find out first.
First Contact With the Grox: The first Grox planet I flew to did not send any ships, maybe I was lucky I don't know, but I was able to talk to them and do the 10 gifts of 100k and get my first mission. Funny thing was they wanted a plant sample from the planet I'd colonized close to them and since I'd terraformed it from T0 to T3 I already had it in my cargo hold. I didn't realized this until after I'd flown back to my colony. So I went back to turn in the mission and then I was surrounded by Grox ships, but they didn't attack. I flew down to the planet and the Grox ships swarmed around me, but still didn't attack.
It was either after I turned in the mission or flew to the next Grox planet for another that the Grox ships started attacking me and wouldn't stop until I finally allied with them. Sorry I don't recall exactly when they started attacking me, but I reloaded the game several times trying to figure out how not to get attacked with no success. So expect to be constantly attacked by the Grox while you are doing missions for them until you get an alliance.
It would be great to find out if there is anyway to do Grox missions without them attacking constantly. I hope someone figures that out and posts back here how to do it. As it is one should be able to do the missions with a boat load of cash and repair packs. The Grox do pay very well for their missions too, I made 1.2 million on one, so maybe not that much cash is needed. While the Grox do not ever trade with you they will repair and recharge your ships for cash like any other civ.
This brings me to an important point: ALWAYS SAVE YOUR GAME BEFORE DOING ANYTHING WITH THE GROX. And another important point: TURN OFF YOUR AUTO BLASTER BEFORE THEIR SHIPS START ATTACKING YOU. You do not want any negative points with them for attacking their ships even though they are attacking you.
Strangely, most of the missions they gave me involved going to my own planets. I failed at least one mission which lowered my relationship score enough that I had to reload. One mission involved gathering up artifacts and delivering them to my own colony lol.
Oh yeah, make sure to start trade routes with them asap. You get positive relationship points for each of the 3 routes you can have. Plus you get to buy the system eventually. In my game they are happy with 1 million sporebucks for a system, but if you pay less they will go to war with you for being cheap. Did I mention SAVE OFTEN? I believe you can start trade routes as soon as they show a blue happy face, but I'm not entirely sure if sooner isn't possible. It would be helpful if someone could check on that.
Finally I got the green happy face with them and formed an alliance. They sent me their "Fist of God" spaceship which indeed makes all this trouble worth the effort.
When I do this again with a non-cheat game I will go ahead and head for the center of the galaxy at this point because it will make the next steps I took a lot easier with the "Staff of Life." Also in my game the Grox are the only civ close to the center making it easy to planet hop there. As it was I did manage to get there later with only Interstellar Drive 4. I don't know if one could make it with a lesser drive, but I don't think so as the distance you can travel become quite a bit less as you get closer to the core.
Next I wanted to see if I could patch up my relationships with my former allies, at least the lvl 5 ones so I could get their ships too. So I headed back to where I started. Most everyone had declared war on me, but at least one was just upset. So I started by bribing them, but the best thing to improve relationships is to terraform their planets. Just taking one of their planets from T0 to T3 made them happy enough to re-form an alliance. This is where the "Staff of Life" would be most useful as one could simply terra-bomb a planet into friendliness without the "Fist of God" making things worse by destroying their ships while you are terraforming the usual way.
I continued re-forming alliances until I had 4 lvl 5 ships plus the Grox "Fist of God" ship with me. Now I'm essentially invincible; I can go anywhere and no civ can hurt me much at all before they die or are terraformed into a friendly relationship.
Props to TheAngelofDeath for finding what I believe is the ultimate strategy