Upgrade Options for Shimano Crankset Tourney FC-TY701
What you have is a basic 3-speed square taper crank arm built for an 8 speed chain. The important parts to consider when replacing it is the BB interface (square taper) and the chainring spacing (8 speed). If you don't want to upgrade the bottom bracket, then you will have to get another square taper crank arm. 

The chainring spacing is less strict, there are differences between 8, 9, and 10 speed crank arms in the chainring spacing. You can be off by a gear or two but the shifting will be a bit less than optimal. Here's some more discussion about it. Personally I have no problem using a 10 speed crankset with an 8 speed chain/cassette on my mountain bike, but I've also been riding it long enough to be sensitive/responsive to mis-shifts that might happen because of it. I consider it an upgrade, as the 10 speed crankset is quite a bit lighter and stiffer, but if you are prioritizing reliability over eeking a bit more efficiency out of the bike, I'd stick with an 8-speed crankset. 

FC-M3000 This is meant for 6/7/8 speed drivetrains and is square-taper, so it would be an exact replacement and you wouldn't have any issues or have to upgrade anything else, but it's not much of an upgrade. 

FC-M430 Alivio This is a square taper 3x9 crankset, so it would work even though it is not specifically 8-speed, and looks like a solid upgrade from the most basic model. 

FC-M610-L Deore This is a nice crankset. Also 2-piece BB and 10-speed spacing. I have one like it on my bike. It would work with a new BB but would be sub-optimal shifting with 8 speed. FC-MT500-3 is basically the same. 

You don't need to stick with Shimano either, something like the SRAM S600 would work also, as it is 8 speed and square taper.