Hey @Angelica_Isacson,
welcome to the community 
I definitely think Asana would be the right tool to go with and over time you will see how much more efficient everything is.
Now onboarding and transferring to a new system is always a bit challenging (at least I have experienced this with various teams) however over time once everybody sees how much more efficient the workflow is they will start to love it more and more.
Depending on your team size I definitely recommend having at least 1-2 coordinators become masters of Asana so they can always assist, guide and monitor.
I recommend starting by setting up the initial structure, then play around with it, test things to see how things will work out best for you. Have a look also at all the guides provided by Asana, they are really helpful: https://asana.com/guide and whenever struggling with something there are many in this forum there to provide tips 
Now regarding handling client projects, this is a tricky question and I am sure there are many opinions about it on how much you want them to interfere or not.
I can tell you how we sorted that over the years:
For one of our companies, we created a team in Asana for every new client we onboard.
Then within that team we have one dashboard for our coordinators (where all tasks are listed)
Then we have a separate project for the marketing team, the developer team and the design team where only the relevant tasks for these team members are added. (All the tasks in the special projects are always tagged to the coordinator project as well as they would want to have an overview of everything).
In the very start, we gave our clients comment only access to some projects, such as the marketing project, however, we soon noticed that they got way more involved than we wanted them + this way they were also able to immediately see whenever a task is delayed or communication between our team members regarding the execution of the tasks. Sometimes also our team members added tasks in the project that the client should not see.
So considering all these factors in the end we decided it was not wise to invite the clients to these projects directly
Instead, we created one project within the team restricted to the coordinators and the client where the coordinators then picked specific tasks or updates to be shown there.
Or let’s say the marketing team would require details from the client, then they would create a subtask under their task and only this subtask gets added to the client project.
Then instead of the client seeing every tiny detail of the process, we provided them with regular updates on how everything is going 1-2 times per week via the Asana project status update and they did communicate via Asana Messages + 1 call per week.
Now for another company, the projects we are working on are smaller so we did not create a team per project but instead just one Asana project per client (to which only the internal team has access to) and we would provide the clients with a weekly report (in form of Asana dashboard/Portfolio)
Then we also have some clients with which we built a better relationship over years already. For those, we know that it would be okay to give them more access directly. But always comment only as otherwise, they would just edit out tasks haha.
So yeah to summarize what I would say is it really depends on the clients you perform work for as to how much access/visibility you want them to have.
I definitely recommend as a first step read more in the Asana forum, the guides and then test out various scenarios to see which one would work best for you
Here are some threads I recommend taking a look at as well:
and many more
Happy to assist if you have any questions