Can my laptop run Sierra or High Sierra?

Grant is correct. The direct link to el Capitan in the Mac App Store is in the support document I provided above. It will open the Mac App Store to the El Capitan entry.

High Sierra, has better performance than El Capitan, so should likely work as well or very close to how Snow Leopard does now.

You can of course, upgrade the RAM on that model of MBP for better performance.

MacBook Pro Memory Upgrade Kits (2008-2010) - macsales.com

It can technically update to both.

Neither Sierra nor High Sierra should pose any issue to it. Assuming the Mac is running o.k right now and has no issues, High Sierra should work fine.

You will however have to update to El Capitan first. You will not be able to go directly to High Sierra from Snow Leopard.

See here: How to upgrade to macOS High Sierra - Apple Support

and here: How to upgrade to OS X El Capitan - Apple Support

Do note, that many Apps running on Snow Leopard may no longer work on High Sierra.

Check App Compatibility with High Sierra - Roaring Apps.com

Make sure you backup before updating in case something goes wrong.

How to back up your Mac - Official Apple Support

MacOS El Capitan and later seems to need 6GB or more real RAM to avoid getting in a Paging Rut and simulating a lot of RAM on the drive, just for "ordinary" Browsing and email. That is the main reason users have seen for "My Mac is too slow".

If you have a Rotating Magnetic drive, installing a same-size or larger SSD drive (and replacing the cable as preventive maintenance) gets users saying, "It's like getting a whole new computer!"

2) how come Apple Store on this machine isn't showing [Sierra, High Sierra] as an option?

10.6.8 is so old, that as Phil points out, you have to take an intermediate step and install 10.11 El Capitan first.

The more modern Mac App Store in 10.11 El Capitan can see the later upgrades, and then you will be able to Install if you wish.

17-in MacBook pro 2010 model can run High Sierra, but NOT Mojave.

El Capitan is not available 'as new' to the general public. You need to use the specific link Phlil Provided.