As an inquirer, nothing is expected of you. I asked my priest while I was still an inquirer if I could venerate the icons, and he said yes, but otherwise I didn't participate beyond making the sign of the cross and making prostrations along with everyone else.
For catechumens, at my parish, there's a weekly class on Saturdays where we meet with the priest before Vespers. He teaches a lesson and is available throughout the week to talk to for guidance or to go more in-depth on questions we have about the lesson if we run out of time on Saturday. Most people seem to spend about 9 months as catechumens, but from what I hear that varies from one priest to another.
My priest also set me up with a prayer book and a daily prayer rule when I became a catechumen, as well as Scripture to read. He's guided me on getting icons, reading Scripture prayerfully, fasting, navigating conversations with Protestant family members, and much more. Also, since becoming a catechumen, I've gotten more involved with the church, helping with meals and other tasks.
Becoming a catechumen involves a small rite prior to the start of Divine Liturgy where the priest asks you formally if you want to become a catechumen. After that, you're added to the prayers for catechumens, and as my priest has explained it to me, you are basically a member of the parish for all intents and purposes other than taking communion.