Estimated trophy difficulty: 3/10 (my rating, will use the proper difficulty once it gets more votes here: Trophy Difficulty Rating)
Trophy breakdown: 0 1 4 7
Offline trophies: 8 - Master of the Universe, Party at Sky Vault, Thunder, thunder, thunder!, Awesomenauts, roll out!, Bionic family, Hawk, wolf, puma and bear, T-U-R-R-E-T powah!, Wings of Silver
Online trophies: 4 - Let our powers combine!, In pole position, Slimer's evil twin and Yoooo JOE!
Approximate time to 100%: 10 to 20 hours, greatly depending on skill or boosting partners.
Miss-able trophies: None
Step 1: Complete the tutorial.
Sadly, the tutorial is about as close as you're going to get to a story mode in Awesomenauts. It has you playing as Sheriff Lonestar, one of the 6 (hopefully, soon to be 7) Awesomenauts available to choose from taking orders from who I'm going to assume is Slimer. He briefs you on the basics of the game, such as killing neutral creeps for Solar and health, buying and using upgrades, etc.
Step 2: Start playing!
There's really not much to this game. You either play online or offline, against bots or other players. You start off with 3 of the 6 Awesomenauts unlocked (Froggy G, Sheriff Lonestar and Leon Chameleon) and you unlock the others, along with more upgrades, levels, and play modes as you level up.
[PS3T Would Like to Thank Nave for this Roadmap]