I do not think there is any difference between road and mountain bike cassettes except that mountain bikes will generally have a wider range.

If a 28 front and 32 rear can't get you up a steep bridge, swap out the front 28 for a 24 but I would wait to see if it is needed before buying a different chain ring. The 28 probably shifts smoother than a 24 would. I have a 24 front granny gear but shifting up off of it is not a smooth shift.

I do not have experience with the Tiagra but I assumed a long cage would handle a 32 just fine. If the bike came new with a non-stock cassette that the dealer may have swapped, you might want to double check that the derailleur is the long cage that the bike should have had new. Maybe the dealer swapped that for another too?

If the bike came from the factory with an 11/32 and the dealer changed the cassette, maybe the existing chain is long enough?