Like the Soldier, Demoman excels at crowd control with his wide-area explosives and doesn't do much else. They have different secondary strengths, however. Where Soldier has team buffs, the Demoman's spike damage with sticky bombs is a good way to eliminate Uber Medics without popping their charge and makes for high initial burst on tank spawns. Demoman is also one of the only two classes (Spy) that gets more than attack speed melee upgrades, making hybrid Demoknight loadouts viable.
Same upgrades as Soldier, focusing mostly on damage with maybe a smidgen of resists just to be safe. If you're looking to Demoknight, you'll want Crit-on-kill early though not necessarily on the first wave (unless it's an Endurance mission). You may be better served with resistance to the bulk of the first wave, or attack speed and health-on-kill with your starting money.
Primary: Grenade Launcher, Ali Baba's Wee Booties
Stick with Stock over LnL here. The higher single-shot burst isn't particularly important in MvM. The larger clip gives Stock better damage over time before you can afford upgrades and keeps it competitive afterward. Not having to direct hit is also useful when firing into crowds as opposed to easy-to-hit giants, since any miss cripples LnL's crowd control capabilities. Booties should only be used if you're upgrading the Sticky Launcher, in which case you'd probably never use your primary anyway and may as well take the 25 free health.
Secondary: Sticky Launcher, Scottish Resistance, Chargin' Targe
Sticky Launcher is better as a dedicated weapon as the faster priming speed and being able to explode a widespread carpet with one click (and without looking) is important when using this as your main weapon. ScoRes is better if you're maining the GL, as you'll mostly only be setting stickies pre-round for spike damage on the first Tank or Giant (14 > 8) and instagibbing Uber Medics.
If you don't need to worry about Uber Medics, then you don't strictly need a sticky launcher. It's perfectly reasonable to rely on primary weapon fire and you can just use the Targe for the charge and substantial free resists. You also need to equip it to access the Damage and Crit-on-kill melee upgrades, though I think you can buy those upgrades and they'll stick even if you switch back to a sticky launcher (even if you can no longer see them).
Melee: Bottle, Eyelander, Half-Zatoichi
Incidentally, any of these are useful if you're not Demoknighting. You shouldn't be using your melee weapon much anyway, so the HZ has extra upside over Stock that you should consider: HZ is a free health refill with smart use, but if you miss...) While the Eyelander has downside, if you don't die you get a substantial amount of free health and movespeed after picking off a few supports or stray Scouts.
Melee upgrades are useful, however, if you want to invest in them. Melee damage is far cheaper to upgrade and the crit-on-kill will persist through weapon switching, so it's useful even to standard Demomen. The HZ will continually keep you topped off as you kill bots, which is really easy to do once you start critting, but the Eyelander is highly appealing for the mass amount of extra movespeed and health you get once powered up. While its healing is nowhere near that of the HZ, you can combine it with health-on-kill since it does overheal. The value of extra health and movespeed is more than not having to buy health-on-kill, however recollecting heads if you die is a pain.