First week and month as a scrum master

I like to start with a sailboat retro to get a big picture. Out of that, a lot of minor topics come out. I also do a knowledge map of what areas of know how do we need and what do we have. Having single source of knowledge carrier is a core problem and I tend to address this early. This helps to get a focus on the value and on working together what many impediments addresses.

that scrum master decides how to get the work done

Think about that! Read again the Scrum guide and argue with the Scrum guide. The Scrum Master is not a Developer Lead. Of course you can support and guide if the team needs it, but definitly you should not decide. Probably your organization has to learn in this case. Ask why the company 'choose' that model, what are the reasons for this setup and change it.

What is the scrum master's role in a release?

Support the team to deliver! How was this handled in the company before Scrum? Again, get the history, ask what are the expectations of Scrum/Agile in the company. 

The first step should be to make sure that everyone - yourself included - understands and agrees upon what the Scrum Master's role is. The Scrum Master does not "oversee" teams or "decide how to get the work done". Scrum Teams are self-organizing and self-managing, which means that the people doing the work decide what to work on and how to go about getting it done. The Scrum Master can, depending on their background, knowledge, and experience, offer coaching into various practices that can promote agility and Scrum, but should not be deciding, dictating, or overseeing the work.

From there, I like to spend a lot of time observing. I'd spend at least a Sprint mostly watching the current practices. I may ask some questions to learn more about what the team is doing and why they are doing it. Depending on the number of teams and the timing of the different events, it may take more than 1 Sprint to be able to observe all the teams carrying out all of the events. Once I begin to understand the current state, I'd want to learn about the biggest impediments and pain that the team feels are stopping them from being effective and figuring out how to solve those. Depending on how big those problems are, it may also be a good idea to look for some quick wins to demonstrate the value of the Scrum Master in resolving impediments.

Specifically in the context of releases, that depends on the organization, but I'd rather see a coaching and facilitation aspect there. The group of Developers should have all of the knowledge and skills needed to produce a done Increment, so they should ultimately be responsible for the release. As a self-organizing and self-managing team, they can decide how to go about structuring the work for a release. If there are dependencies or complex interactions, the Scrum Master would be able to facilitate to make sure that the process of releasing the product is smooth and there aren't any roadblocks.

I have joined an organization as a Scrum Master. My manager is an IT delivery director.

A Scrum Master is a servant leader. Your manager is the team, not a higher-up with a narrow organizational function who thinks they direct delivery.

They are looking for someone to come in and solidify their Agile processes. As a scrum master, I will be overseeing  two teams.

No, as a Scrum Master, you've had the buck for cultural change passed on to you. That's what's happened.

Rather than "overseeing", which requires no corporate change, it might be better to look ahead ​​, guide, and facilitate for those teams. A Scrum Master is a more effective servant leader than an overseer, and the best way to solidify such practices in an organization is to exemplify them to others.

The teams and the whole company truly embody the Agile values. They have an excellent culture! 

That's good. What evidence has convinced you of this?

I do have a couple of questions for the community. When I asked a product owner about the scrum master's role in the teams, he told me that scrum master decides how to get the work done. He stated that the product owner provides the team the "what'' needs to be done. Scrum Master decides and helps the team decide how to get the work done. 

A Scrum Master manages people's understanding of Scrum, including its artifacts, events, rules, and accountabilities. Do you think that the Product Owner's beliefs and assumptions are consistent with the Scrum Guide? Might there be an opportunity to improve his understanding, and therefore a call to action for you?

What is the scrum master's role in a release? I understand that the Scrum Master should facilitate an environment for the team to deliver successfully.

Correct.

But is the scrum master responsible for the release?

The Developers are responsible for doing the work needed for a release to happen. They are therefore accountable for its quality, while the Product Owner is accountable for maximizing its value.

When the team succeeds the Scrum Master succeeds, when they fail, you fail.

Finally, I would love to get some advice on what would be great to start as a new scrum master in a new organization. I have started having 1:1's with my team members. 

Find out what issues are being squirrelled away in those 1:1's which impede a consensus view, amongst all team members, of what is going on. It will be difficult for your team to collaborate and to inspect and adapt effectively without it.

Finally, is it good to ask for a list of resources and good contacts to have in order to help resolve or connect the teams to resolve any impediments they may have?

What you are looking for rarely comes in a list, it is more likely to come in relationships. The most wicked constraints and impediments are invariably organizational ones, so find out who your sponsor is. Find out who among the higher-ups cares about agile outcomes and is empowered to overcome organizational gravity, and keep them close.