Your bike has a triple crankset and a 9 speed cassette.
That's pretty much all the details you need to match in your new shifters.
What style you choose is personal preference, but they have to suit your flat-bars, so no road-bike brake/shifters.
I personally prefer to keep components separate where possible, so I would avoid flat-bar shifters that have integrated brake levers. Sure they save space on your handlebars, but break one and you have more to replace.
Do take the opportunity to fit new inner gear cables; it is barely any more work and a nice smooth end is easier than reusing the old ones. Evaluate the outers too - now's the time if they need doing.
Your bike is not new - it could be a waste of money to drop new high end shifters onto it. Any of the middle groupsets would be more than adequate, like alivio and altus and acera. Tourney would work fine too despite being the "lowest" groupset. You could put Deore on, for completeness if nothing else, but XT or XTR or the other high end groupsets would be a bit pointless.
Do note that GRX will not work, that's a newer groupset that meshes road and MTB values and likely won't shift well if at all and doesn't have a 9 speed.
Check out Ebay or craigslist or your local recycling outlet - its possible you could get an entire donor bike for less than the cost of new shifters.
Another option is to go to a 10 speed groupset, if the costs work out for you. There's more to replace, so it might be too expensive unless you have a donor bike.
- Rear Derailleur
- Right hand shifter for 10 speed
- Left hand shifter for triple crankset.
You should be able to leave the front derailleur alone. The existing hub on the wheel will take a 10 speed cassette unmodified.