This bike (I worked on a Tourney equipped bike over the last few weeks for a friend which did indeed have a 7speed freewheel rather than cassette+freehub) probably has a 126mm spacing between the rear dropouts which isn’t compatible with 8speed or higher casettes even if you change the wheel & therefore hub. You can stretch it in some cases but likely not to the 135mm you’d need. If you need more gears to get the 11-32 range you want which I think you would. And you also need to change the shifters of course in that case.

You do need a special splined tool to get that freewheel off, and to work with a cassette you need the same splined tool to loosen and tighten the lockring plus a chain whip to stop the freehub spinning, just when you try to loosen it.