So, any vacuum will work, even SOME uprights. However, any vacuum you use needs to have the ability to turn off the beater bar(the rotating bar for carpet). Otherwise you'll be kicking stuff out behind the vacuum and potentially scratching the floors.
As for moping:
Get a Swifter, toss out the wasteful pads that come with it. Get some microfiber cleaning cloths, they work great on Swifters. Get a small spray bottle, fill it like 90% full with water then add a small amount vinegar. Rinse the cleaning cloth every like 5 feet and squeeze as much water as you can out of it. Spray the floor with the vinegar solution ..then mop.
ALWAYS work in like 5 foot segments. Cleaning more floor then this will just leave streaks of dirty water.
You could also just use a small squirt of dish soap honestly.
NEVER use Murphy's Oil soap on hardwood floors. The bottle says that is what is is for but, it will leave a residue on the floor over time. The company that makes it has even said in letters to people that you shouldn't use it on floors but..... they still market it that way. It's so annoying.