When multi-classing, do you take the equipment from your new class and add it to the equipment you got from your main class?
No, you don't get the equipment.
The way the multiclassing rules are set up, it provides a "positive" list: a complete list of all the things that do happen are mentioned, and anything not mentioned does not happen.
The multi-classing section (PHB, pp. 163–5) lists these things that you get (with notes on any changes to how they work):
- Hit dice and hit points
- Class features (which has the one equipment exception: you do get a spellbook if necessary)
And that's it. If it's not HD/HP, Proficiencies, or a class feature, it's not included when you take a level in a new class.
This makes sense in-world too: that equipment you get at 1st level represents your character's entire life savings — when you decide to take a crash course in fightering while on the trail, you don't coincidentally find a lifetime's worth of weapons, armour, and earned gold just sitting by the side of the road.*
* If you do coincidentally find a stash of sweet gear by the roadside it's because The DM Is Making Adventure Happen To You and you should start wondering who lost their armour and whether they're going to be back for it soon.
No, you don't.
From PHB pg. 164:
When you gain a new level in a class, you get its features for that level. You don't, however, receive the class's starting equipment...
This clearly states a clear answer to your question: When you gain a new level in a class (because this is under the Multiclassing rules, this implies any level in any class besides your first level in your first class) you do not gain its starting equipment.
As mentioned by @SevenSidedDie, the answer is no.
I think it's also worth noting that the value of starting gear is quite low. The most expensive items would seem to be the Chainmail (75gp) and the Hand Crossbow (75gp) along with possibly an expensive equipment pack.
So if you're multi-classing into a Fighter at 2nd level, then this might be a big deal. But by level 4 or 5, such a windfall would be a pretty small benefit anyway.