Improving the process is not as difficult as you may think. It is more difficult if you don't have extensive knowledge from past companies. It is possible to do a very good job on your first attempt. You'll learn what works for you after you do it the first time.
Here are two books that I recommend you buy:
1. Systematic Software Testing by Rick Craig and Stefan P Jaskiel (substitute any decent software testing book for this one as long it covers the issues in the second book)
2. Test Process Improvement by Tim Koomen and Martin Pol
You are fairly junior at two years (no offense meant). My opinion is that you will need a process that has been in use and can be referenced. I like and have used the above books for new hires and process improvement. Managers can't take your past experience as a guide so you will need to show them where you are getting your information and why they should listen to you. You're expecting them to trust your judgement which may be hard to do without past experience or a defined process. These books will help.
I have lots of experience but I still showed the book to our board when I give a presentation. It gives them a warm fuzzy feeling to know that I am not just making this up as I go along. They know my experience level but it is always a plus to show that you are using a defined plan.
I don't get any kickbacks from these guys. Just like their process. It will really help you to know what to update and in what order if you don't know where to begin. Don't blindly do everything the process says. You should have enough experience to know what items to avoid for your organization.