The big issue that I see, is that the bike most likely has a freewheel, not a cassette, so swapping the rear wheel would be necessary for upgrading to 8 speeds. If it is only going to sit on a trainer, you could probably get a Sunrace 8 speed freewheel, but I wouldn't recommend that for actual riding. They aren't released yet, but Shimano A070 shifters will be 7 speed. There are also discontinued 7 speed shifters that you can probably find too.

When riding the bike, try and imagine what it would be like with drop bars. I was able to guess that they would feel great on my Schwinn, so I did a conversion, and was right. Pretty amazing that the bike felt great with 4" riser bars, and also drop bars 5" lower than the saddle.