Идти is not an auxiliary verb, it refers to literal physical going/walking. Yes, it's also possible to say я шёл гулять, but it has the same meaning of having to get somewhere before the "actual" walk begins.
In fact, I can only think of one auxiliary verb in Russian besides быть. It's стать. On its own, it means "to become"; when followed by an imperfective verb in the infinitive, it means "to begin [to]". That is, it's broadly synonymous with начать, but начать's meaning of "begin" is lexically inherent, whereas стать in this sense is truly auxiliary. You can say, as a complete phrase, Мы уже начали "We've already begun/started [doing smth.]", but *Мы уже стали can only mean "We've already become" and wouldn't make sense when talking about starting something.
As for the nuances of usage, стать implies less of a clear plan of action, or intent to carry it through, or a more general expectation of consequence, than начать.
Вадим стал расхаживать из стороны в сторону.
Вадим начал расхаживать из стороны в сторону.
These are both "Vadim started pacing to and fro", but the difference is, the first one is just a thing that happened; with the second one, you vaguely expect it to lead to something else. Perhaps it's a thing with Vadim that he's known to start pacing to and fro when he gets angry and is about to yell.
Мы стали обзванивать больницы.
Мы начали обзванивать больницы.
"We started calling different hospitals [around the area]", presumably because someone's gone missing. The difference is: the стали one has the implication is that they were unclear what to do or were out of options at that point. Whereas the начали one sounds more like methodically going through a missing-person checklist.
And another important nuance. A negated or interrogative auxiliary стать has a significantly different meaning. In a way, the opposite of regular стать. It doesn't even mean "to begin to" so much as "to decide/choose to", with a particular shade of meaning that I'd describe as objection-based refraining from an already proposed or implied course of action. The objection in question can equally be moral or rational.
Я не стану выманивать деньги у матери троих детей.
"I will not coax money out of a mother of three."
Разве они станут связываться с племянником олигарха?
"Do you really think they'd mess with an oligarch's nephew?"