You begin with a neutral influence on all races beside the one you choose to start with as you stole their experimental flagship. There are solutions to make races that hate you join the federation (skylaxian senate can achieve this miracle), as such you don't really need everyone to like you, you mostly need them to like each other.
Even if races hate each other they will ally in the fight to prevent you from delivering spacefaring tech. For the "my federation", actually it is not your federation, it is their federation
.
Arcutians threw one of their moons at the Hydral homeworld, it's like their signature move, you'll learn to respect that the hard way.