As a long time wargamer:

You want plastic model glue, AKA polystyrene cement. It fuses the parts together. The models themselves will break before the bond will. You don't have to worry about getting the glue on tools, hands etc as much as with other glues as it will simply wipe off of non-plastic items, although you will want to avoid using too much or getting it on parts of the model other than those you are gluing or it may mar the detail. Polystyrene cement doesn't dry out in the tube, so you can put the cap back on and the glue will still be good to use a decade down the road. Pretty sure I have a tube or two of glue in the hobby kit I've had for 20 years that would still work great.

Your other option is superglue, AKA cyanoacrylate or CA glue. It's more expensive and messy to work with, and you run the risk of gluing things unintentionally to other things. If you've not gone with an industrial level superglue the latter is mostly just inconvenient, although there can be occasional discomfort if you glue something to your hand. As superglue dries it "off gasses", and can leave unsightly white patches of dried glue vapor around the joint. It will also off-gas in whatever container it is in, which is why it is usually provided with an additional sealable container to store the glue bottle in or you end up with that white residue on everything in your hobby kit. The joints themselves are also more likely to break, requiring all too frequent repair, especially on things like outsretched arms or other joints with low surface area or higher leverage. Unlike polystyrene cement, you will have to supply constant pressure to the joint as the glue dries or the bond will fail, which is usually when you discover you've used a little too much glue and now you have to peel the model off your fingers. The glue itself tends to dry out in it's container rather quickly because any little bit of spilled glue on the nozzle prevents a proper seal with the cap. This leads to blocked nozzles, stuck caps, or a bottle with a solid lump of glue in it after a couple of months in storage.

For any non-plastic parts or models (resin, pewter etc.) you are stuck with superglue.

TLDR; Polystyrene glue yea, superglue nay.

As far as painting assembled models goes, that depends entirely on the model. If gluing a part on creates tight spaces hard to reach with even a fine-tipped brush, you're much better off painting these pieces separately. If you plan on doing the painting in the near future, you could hold off on gluing these pieces for now. If the painting is one of those "I'll get round to it some day" ideas, trust me, you won't. May as well just glue it all now. As first time model-painters, you're probably not going to be producing stellar quality work in any case, I wouldn't worry about it overmuch.

If you do actually decide to paint them before assembly, be sure and leave the surfaces where glue would be applied free of paint. You can touch up any missed areas after assembly.