Might be shorter to list what you can keep:
- Frame and fork(probably, if the OLD spacing is the same)
- Front wheel
- Bars and stem
- Saddle and seatpost
You will need to buy or source replacement:
- Rear wheel hub because a freewheel and a cassette are too different. And its a sad truth that a whole-new wheel might be cheaper than a hub and rebuild onto your existing rim, even if spokes can be reused.
- Cassette, more cogs.
- Chain, the thicker 7 speed one won't go reliably onto the 9 speed cassette because they're closer together, by a little.
- Rear derailleur
- Right hand shifter, it needs 9 positions and therefore 8 clicks.
You should get away with leaving the front chainrings completely as-is, with the shimano shifter and front derailleur. As long as you're okay with visually-different shifters.
I've personally changed a 3x5 bike to run 3x9 deore and made no replacements to the chainrings or cranks, though I did have to modify the cage of the front mech to be a closer fit. And I have friction shifting on the front, for fine control across the range of gears.
If that doesn't work, you may also need
- Front shifter
- Front mech
And then you might find the spacing between the chainrings has changed which will require
- Replacement spider
- Possibly replacement chainrings to fit that spider.
- Possibly replcement cranks too, if they're integrated with the whole mounting system.
Do make sure the derailleur is rated for the cassette you intend to fit, AND that calculation includes the difference in tooth count between your large and small chainrings.
Reading between the lines you have a 3x7 bike with Tourney "groupset" so its unlikely to be a high or medium range bike. Tourney was the bottom shimano branded groupset, and while it does work, it is cheap.
So one might presume the rest of the bike is of similar specification and quality.
Be careful about throwing a lot of money at a bike in the form of upgrades. You might find that for "similar money" you can have a whole new bike with all the same components. This also leaves you with two bikes, not one and some scrap parts.
The 3x5 to 3x9 upgrade I did used an entire groupset salvaged from a bike that had suffered frame damage by being crashed and then driven over, but the car's wheels avoided the transmission. So if the parts are free then its feasible, but at new price it would have made no sense.
Those shimano shifters have integrated brake levers with long cable pull for V brakes. If you choose an integrated shifter/brake from SRAM then it will also have to be long-pull for your V brakes or cable-operated disks, whatever you have now.
You can also elect to fit separate shifters and brake levers instead of integrated ones.