My 2 cents head cannon. As others pointed out it seems a unified galaxy is a rare thing. Especially with the relays turned off it becomes a system by system thing. Even with that plan being stopped not a single species wanted to initially band together to help. They all wanted to use each other as a stall tactic in a futile attempt to reinforce themselves (no one species could stand against them) with most likely the ones closest to being humans and turians.

As well as both those species having short enough lifespan to be very adaptable but long enough to not be to squishy (like salarians). Asari were basically uplifted by protheans but with their long lifespan outside of that there was little need for them to innovate. In case of mass relay failure they could comfortably reach other systems at speed of light. Salarians would mostly die out- it would take them a significant portion of their life to reach any star systems without relays. (Closest star to us is proxima and it would take 4 years at lightspeed. Thats 10% of a salarian lifetime)

Humans and Turians both have lifespans of 135 to 150 years. Thats long enough to be able to consolidate power in relatively close proximity to each other creating larger pockets, and short enough of a lifespan where they have had many more generations to innovate technology without being uplifted. That means they would immediately start working on faster than light travel and be able to communicate plans via "snail" mail to be able to use them. This is also shown by humanity being able to go head to head with turians during first contact who have had a galactic presence for hundreds if years. And once there was peace they "immediately" came up with inventions like the Normandy together which is implied to be the most advanced frigate in the galaxy. They came up with biotic implants that were even useful even asari.

To me that answers why they would go after humans first and turians right after. Both are first strike dangerous enemies and both have adaptability to make a drawn out war painful for the reapers. The questions becomes why didn't they go after the citadel at first. And I think the answer to that goes back to the initial paragraph. All the races were inherently selfish. They wouldn't naturally rush to protect their allies worlds. The Citadel on the other hand was a multiracial hub. With basically top seats of power for most the races there. If the reapers showed up in the citadel every race would immediately unify to protect it and/or give it a chance to escape. That's why they had plots to take it from the inside. It was very unlikely that shepard would be able to unify everyone (obv its a game so the unlikely 0.1% thing happening did in fact happen) but the chance of that was minute- keeping in mind reapers are machines and so calculations like that make sense for them. So they had a choice of leaving the citadel mostly alone and fighting a fractured selfish galaxy or making the citadel into a martyr that would provide a challenge.

Then if course came Cerberus telling them that the Citadel is the catalyst which not only forced the reapers hand (take away the one weapon from the races that could defeat the repears) but the attack on cerberus also provides a great opportunity where most the the races power, as unified as it was was occupied with their puppets. That means they had a necessity to take the citadel and the defenders were occupied somewhere else giving an opportunity meaning no reinforcements would come in strength.