Upgrade cranks, chain, gears on a Carrera Vengeance MTB I have a Halfords Carrera Vengeance 2011/12 bike:

Carrera Vengeance Mountain Bike 2011/2012
The Carrera Vengeance  Mountain Bike will have you powering up steep hills and across rugged trails in no time. It has a Suntour XCM-V3 suspension fork which soaks up bumps like a sponge, and its SRAM 24-speed gears will make the afore-mentioned hills a piece of cake. 

Approximate Weight (KG): 14.6kg 

Forks: Suntour XCM-V3 

Frame Material: Alloy 

Front Brake: Tektro IO Mechanical disc brake 

Number of Gears: 24 

Rear Brake: Tektro IO Mechanical disc brake 

Wheel Size: 26" 

Cassette: 11-32T 

Chainset: SRAM S200 3.0 chainset 

Fork Lock-out: No 

Fork travel: 100mm 

Frame Colour: Satin Gunmetal 

Front Mech: Shimano FD-M190 

Gear Shifters: SRAM X4 trigger shifters 

Gender: Mens 

Handle Bars: Alloy oversize riser bars (30mm rise) (31.8mm diameter) 

Headset: Semi integrated 

Hubs: Alloy with Q/R 

Pedals: Wellgo alloy platform 

Quick Release Front Wheel: Yes 

Quick Release Rear Wheel: Yes 

Rear Mech: SRAM X4 rear mech 

Rims: Double wall alloy rims 

Saddle: Carrera 

Seatpost: 350mm Alloy 31.6 

Stem: Ahead aluminium stem 

Suspension: Front 

Tyres: Kenda Tyres

But the gears are quite bad IMHO. It takes a lot of force to switch gears. I'd also prefer bigger cranks as I pedal on roads a lot. I'm perfectly happy with the brakes and the fork.

I'm looking to buying a set online, e.g. from https://www.merlincycles.com/groupsets-81168/ , but they come in a million sizes. How do I find one that will fit my bike?
ANSWER This is a bad idea.

Your bike has 24 gears total, so its probably a 3x8.  Thats three front chainrings and 8 cogs on the rear.

You could get more road speed by having a smaller small on the back, or a larger large on the front.  However the default rear cassette is 11-32 and 11 is as small as they go.  Your specs don't say what the front chainrings are, but on a triple MTB they're likely 26/36/46 tooth, or similar.   If you put a 50 tooth front chainring on, then the FD has to be able to get there.  Most simply can't handle more teeth.

If I were you I'd look at the gear cables and see that they're moving okay through the outers.  The inner wires should be shiny and slide easily.  A new set of inner/outers will cost you a lot less than a group and should help sort your shifting.  Also measure your chain for wear (ie stretch) and if its over 1% elongated, replace it and the rear cassette if it is worn or starts skipping.  

You ride on the road a lot, perhaps you could consider a dedicated road bike?  If you've got the budget to buy a new groupset, you're within spitting distance of a complete road bike anyway.  And then your N is 2.

Summary: You're on a hiding to nothing doing expensive part upgrades on almost any bike.