I thought dipping into Barbarian would make my character a lot faster. I don't know why, but he's not moving noticably faster than any of my other characters.

Do they actually move faster?

Yes. About 15%. It's like taking Dash as a feat. However, iirc the barbarian's elevated movement does not work if it is wearing heavy armour.

If you really want to test it out, set up a race between your barbarian and a character that does not have a boost to speed (e.g. not a barb, not a monk, does not have Dash). The barb will be faster.

The fastest character is a high level monk, as monks receive a movement boost every three levels.

Okay, I'll test it out. BTW, my Barb is holding a large shield(NOT a tower shield), does that matter?

Which is faster, a 3rd level Monk, or a Barbarian?

Shields shouldn't matter, as barbs have the shield proficiency/feat automatically.

From observation, I think a barb and a monk have about the same speed as 1st lvl characters. A lvl 3 monk should be a bit faster.

I'll set up a race then, as you said. Maybe he is faster, just not too much.

Is a level 1 Monk any faster than any other level 1 (non-Barbarian) characters?

I believe so. It doesn't say so in any of the character descriptions I've looked at, but I have the sense a monk has better-than-average speed. I might be wrong, though. ;)

Just to add: even if you do take the heavy armor proficiency for the barbarian, he will still not get fast speed when wearing heavy armor.

Personally for me it was a blessing. I find it a headache when one character moves ahead of the rest of the party. High speed should be an ability that can be turned on or off.