I have a few questions: How do you level up weapons, how do you level up your character, and how do you unlock new weapons? Thanks.

You don't level up weapons or characters in TF2. Some items state what level they are but as of this moment in time TF2 has no leveling process. I guess Valve are just playing around and adding to the overall comedic / cartoon vibe the game has.

To unlock the new weapons you need to have unlocked the Milestone 1, 2, 3 achievements for the classes. However to unlock each Milestone achievement you have to have gained a certain number of achievements and you can see how many you need by simply looking at the Milestone 1, 2, 3 achievement's description.

In short you need to gain achievements to unlock the items. How you do this is up to you.

When I get killed sometimes it says there's a bonus to the items (like lvl 10 scattergun has +20% clip size or something). Do I get that by unlocking the milestones?

"When I get killed sometimes it says there's a bonus to the items (like lvl 10 scattergun has +20% clip size or something). Do I get that by unlocking the milestones?
"

I just got my Soldier up to lvl 70. I wanna unlock his triple rocket jump to divert some aggro away from medics.

Seriously though, Hamz already explained everything. Also, I require your steam name so I can endlessly pester you into playing TF2.

My Heavy is a level 70, he has a shield that can be really useful when no medics can heal you.

/sarcasm

my medic has lifebloom