Any of those answers are fine, although (1) can be ambiguous and (2) is less general with the timing.
In colloquial American English "in X" can sometimes be taken as "in the business/field of X". That is, you may see "in higher education" describing people/events who are not actively studying for a degree, because they're related to the broader field of higher education (professors, administrators, etc.) Not a huge concern here, but something you might run into.
For "go on to higher education", this would be strange to describe someone who is actively studying. At age 18, I was going on to higher education. In my biography, it might say "after getting a scholarship, he went on to higher education." But for someone currently studying, they're not going anywhere at all; they're already at higher education!
And, as always, be aware that terms for various levels of education differ around the world and across levels of prestige. Training to be a plumber, lawyer, medical doctor, engineer, and/or historian might fall under tertiary, higher, vocational, higher, continuing, or other such "education" type terms, depending on where you're standing and who you're talking to.