I need to replace the chainset on my mountain bike
You haven't said why, which might help you/us.
Make sure the inter ring spacing is the same. 9 speed 10 speed etc.
The chainset does not have to be speed specific (eg 7 speed, 8 speed): one either side is fine so do not heed this advice. The OP's Apollo Evade has 21 gears so is a '7 speed' system
you may need to change the bottom bracket
'May do' . Mitigation: Don't get a new/secondhand crankset that needs a different axle length.
the jockey wheels look to be quite worn as well. One is pointy and the other looks like shark fins. Could this cause the chain to slip when moving off from stationary?
This is unlikely to be what's causing the slipping (skating).
the thing that causes slipping when moving off
As
@Tim Hall says, it sounds very like the OP has worn his chain which has elongated, thus also wearing his cassette which results in the chain skating over the rear sprockets when the OP puts the power down eg from a standing start (uphill?). The jockey wheels are innocent OK!
Looking at the cassette there doesn't seem to be any visable wear on the teeth
You can't tell by looking that the cassette is worn. When you experience skating (which you are) you know it's worn.
Unless there is a good reason to get a new chainset (see my first point above), spend your money on a new cassette with an 11 tooth smallest sprocket, if you desire more 'speed'. This is much more economic (especially as you're going to have to anyway, or put up with skating which will sooner or later result in 'saddle/contact point damage'). Dropping from a 12t to an 11t will give you 8.3% more speed in your top gear (at the same cadence). Changing from 44t to a 48t large chainring will give you 9.1%. This links to a 7 speed
11-28 cassette (for your Apollo Evade)(£12) and a chain will cost maybe £12 [less £5 OFF CODE = ECWJ2R *Minimum spend of £30.] I will guess that you'll need a bike shop to fit these, so go along and ask.