My friend and I will fly to Frankfurt Hahn airport and we need to go to Frankfurt main HBF, because we have to go to Bochum. What is the cheapest way to go from Frankurt Hahn to Frankfurt HBf and from Frankfurt Hbf to Bochum?
It is not necessary to travel through Frankfurt. You can take a bus from Hahn to Koblenz and catch a train there. Many northbound trains from Frankfurt to Köln and onwards go through Koblenz. And the fare from Hahn to Koblenz is a bit lower: 11.20 EUR vs. 15 EUR (or 14 EUR if you buy the tickets online).
The full fare for a second class train ticket for 2 persons from Koblenz to Bochum is 84 EUR (or 88 EUR if an ICE train is involved). It can be much cheaper if you book the tickets in advance and commit to specific trains. If you choose to travel on regional trains (RE, RB and S-Bahn), you can buy a Quer-durchs-Land Ticket for 52 EUR. In that case you don't have to book in advance or commit to trains. It's fully flexible and not much longer than a trip on an IC or ICE train.
As has been pointed out, going through Frankfurt (Main) may not be necessary. Busradar gives a good overview of your options for bus and rail travel (they even include blablacar).
In case you have to go to Frankfurt (Main) for some reason, your best guess is the 14€ bus, I'm not aware of any alternatives.