I have quite a lot of weapons to choose from for myself, but how do I give some to my squad? Currently they only have one each, and some empty slots.
Like in Mass Effect 2, squadmates can only use certain types of weapons. When you depart for a mission, you can click on the empty blue slots in order to pick a new weapon for them by clicking on it. Grey slots are weapon types that you are unable to equip for that squadmate. Unlocking a new weapon will allow you to equip it on a squaddie only if they can equip that weapon type.
In addition, when you pick up a new weapon during a mission, if you choose to go to the loadout screen, you can equip your squaddies there as well.
Here you can see that James can use Assault Rifles and while he does not have a Shotgun equipped, he can equip one. He is unable to use Sniper Rifles, SMGs, or Heavy Pistols.
I'm a little unsure on how you've managed only one weapon type for your squadmates though, because I get a popup message saying "You must have at least 2 weapons equipped" if I try to leave the loadout screen with a setup like the above one.
You can't. Unlike Shepard, they can only use 2 predetermined types of weapons. The grey slots on their equipment screen are locked, but you can change what they have in their blue slots in the regular manner. The easiest way to change their loadout is when embarking on a mission, you will get a screen prompting you to choose the weapons for your squad.