Will need to hire a roofer/contractor to fix this problem around the chimney. It's leaking down the side of the house under the siding, and causing wet spots along the walls/ceilings of the 1st and 2nd floor on that side of the house.

I want to be knowledgeable before dealing with any quotes. What is the main problem here and roughly how much should I expect to pay (already have all the shingles and materials). Just so I don't sound too uniformed when they show up.

 

It looks like the sheathing below the roof is gone. The shingles look to be resting directly on the roof rafters.
It's probably leaking down inside the wall too.

The shingles will need to be removed all the way around the chimney and partly up the slope behind the chimney. It may be a tough repair as it looks like your roof may be fairly old and in need of replacement too. It may be tough to splice the new to the old.

Extremely hard to offer a price. I can see a full days work for two guys.
Possibly $1000-1500.

Best to get several estimates.

 

I am most concerned by the buckling and the liberal use of tar around the chimney. It appears water has been getting into this area for a long time. The question for you may be where do you want them to stop? I think the roof sheeting in that area will need to be replaced. You may have other damage like the rafters next to the chimney and the rafter ends may be rotted. Then there is the fascia and soffit. While not technically part of the roof they may have also been damaged by the water. And lets hope the wall below is OK.

Whoever comes to do the work will be in a similar boat. They can guess about the damage but they won't know until they open the area up. It's common to start with a simple "repair the roof" but depending on how much water has been getting in and for how long the damage could extend well below.

 

There is a good possibility there is framing timber damage as well, especially around the chimney. Since neither you nor a contractor can see the extent of the damage, he will want to quote framing and sheathing repair on a time and material basis. I would be wary if someone gave me a fixed price. I have also seen where the homeowner negotiated with the contractor to provide the needed material. It was a disaster because the contractor claimed the owner was the cause of delays in getting the material on time and the owner paid for the delay time, true or claimed. Good luck.

 

I need to mitigated any further damages inside by stopping the water from getting in while I deal with the next steps. What's the best way to tarp this so the water runs off, to buy me some time?

 

You will need to tarp up and over the roof's peak and over the chimney. Basically impossible or very difficult. The thick band of tar you see around the chimney was someone's previous attempt(s) to "buy time". The time for "buying time" has long passed as you are already seeing damage without having to open anything up.