Is there a way to reduce or eliminate the damage taken from rocket jumps?

Since you're new to the game a good source of information is the official wiki.

On the Jumping page there are many good informations about Rocket Jumps:

• you have a 40% resistance when rocket jumping against your rockets that don't damage any other enemy. This is important, since any enemy nearby hit by the blast makes you get full damage from the rocket
• the Gunboats give you a 60% damage reduction. The "no enemies around" restriction still applies
• Crouch-jumping and shot timing have a great impact on the height and length of your jumps
• You can train your rocket-jumping skills with the Rocket Jumper. It completely removes the damage from the rockets, including the self-damage. Pay attention to fall damage though, since it can still kill you.

What you're seeing when you see others happily jumping around may be a combination of all the reasons above. Properly performing rocket jumps always requires skill and the self-damage has always been part of the move, not only in TF2, but in most modern shooters.

You can use the Rocket Jumper to practice rocket jumping without dealing self-damage, however, it also does not deal damage to enemy players at all, making it next to useless for the Soldier Gunmettle Contract Advanced/Bonus Objectives.

I wouldn't worry too much about self-damage. The Soldier has the second-highest Health (after the heavy) at 200HP, so he can take a bit of self-punishment. It also helps to know where health-packs spawn on the map, (and potentially to team up with a Quick-Fix wielding Medic)

I generally prefer other classes, however it only took about 30 mins of practice to get the hang of Rocket Jumping for me. I followed the tutorial here when practicing:

As for completing the contract, here's what I suggest:

Maps: Capture-point with open-air points, or Payload maps. The new CP_Powerhouse's middle point is great for Soldier-spam.

• Primary: Liberty Launcher.
  The increased clip size and blast damage makes clearing objectives/capture points from midair a lot simpler. You'll get a lot more airborne kills clearing these out, especially as the enemy takes damage from your teammates' fire as well, which makes up for the -25% damage.
• Secondary: B.A.S.E. Jumper
  The Advanced part of the Soldier's Contract requires rocket jump kills, and more time spent in the air = more chances at scoring the kills. Paired with the Liberty Launcher, you should have no problem scoring airtime-kills. (I'm usually more of a fan of the Buff Banner for the Team support aspect, but you can't go past the added air-time this will give you)
• Tertiary: Market Gardner or Equalizer
  The Market Gardener deals crits whilst Rocket Jumping, Whereas the Equaliser deals more damage, the more you are injured. The Equaliser is better all-around, but the Market Gardener is also viable.

An answer I haven't seen posted is that there are different types of rocket jumps.

• Straight up: Shoot a rocket straight at your feet. This gives lots of height, and lots of fall damage if you come down.

• 45 degree: Shoot a rocket at an angle towards the ground behind you. This is the typical rocket jump. It gives you good distance, but probably fall damage as well. You might use this jump for jumping onto a group of enemies. You'll have less vertical movement/damage the further out you hit the rocket, so this can give similar (but worse) results to a wall jump.

• Wall: Instead of shooting your rocket at the ground you shoot it at a wall. This gives you the absolute minium damage and vertical travel. It can be tricky to learn, but this is the best jump for getting around the map.

When jumping for speed, the main goal is to reduce the vertical height of the jump. This gives you more distance, and more importantly, doesn't cause you fall damage.

The gunboats and rocket jumper are nice, but it's totally possible to take only ~30 damage if you use the right type of jump for the situation.

A few other tips:

• Use the increased mobility that rocket jumps give you to gather health packs as you go
• Practice! An hour spent on an empty server practicing rocket jumps will really improve your ability.
• Go at it with the rocket jumps when rolling out with a medic. The only class that can beat a rocket jumping soldier to mid is a sticky jumping demo.
• Always crouch when rocket jumping (timing wise, I jump, crouch and fire at the same time, but that can be varied), you get less damage and more speed.
• Binding crouch to a better key than ctrl can make jumping easier. I have it bound to right mouse button for solider

As far as I know, there's a weapon which eliminates the damage but it doesn't do as much damage to enemies as well. Unfortunately, with the others you damage yourself - and this cannot be eliminated.