I just made it to Balmora and I'm already being asked to join the Thieve's Guild. Can I join more than one guild? I'm asking because I was planning on joining the fighter's guild. Thanks for any answers!

You can, but the Fighters and the Thieves do have a conflict, because they eventually turn against each other. There is a loop-hole but it's easy to mess up.
Another reason to not join both of them is that they have very different Favored Attributes and Favored Skills, so advancing in Rank can be difficult.

You can join all 7 Guilds if you want, but it's not recommended. Favored Attributes and Skills are too far spread to be able to gain Rank with any ease. It's also confusing and unbalancing to be in that many factions at once, and makes it very hard to roleplay a character who is friends with all these factions that hate each other.

You can only join one of the three Great Houses per character.

I'd have to go in-game and check myself, but UESP says that they don't have a negative relationship with each other, so I imagine you could join both the Fighters and the Thieves guilds. Advancement might be difficult in one of them depending on how you set up your character, but yes, you should be able to join both.

Thanks guys! If you don't mind, I have one more question. In Oblivion and Skyrim you use the right mouse button to block with your shield but in Morrowind the right mouse button brings up the menu.

I noticed that if I hold down my left mouse button my character holds up the shield. Is that how blocking works?

No.
Blocking is done automatically, based on the level of your Block skill.
The higher your Block skill, the more often you block an attack.