What I'd consider doing is (I've done exactly this, but not specifically in a software project, more infrastructure):
- Write personal notes with a list of problems, broken up into a description of the problem, why it's an issue or a disadvantage to have things in that state, and what a suggested remediation is.
- Go to the project lead / manager / etc. and say, hey, I know I'm new here but I've been noticing a few items that I feel could be improved in order to make the development process faster. I've written my own personal notes on that. Would you be open to / have time to review that if I send that to you? I feel it'd be useful and I'd love to hear your thoughts.
- If yes, then set up a meeting and send them the doc before hand.
That's one approach.