Re: SO far behind–what first?
I always find it helpful to spend a few moments sitting in the garden with pen and paper, looking around and listing everything that needs to be done. Then you can look at your list and prioritize.
Without being there myself and seeing the situation, here are my triage recommendations:
Skip any pruning except cutting out diseased and damaged wood until next pruning season (probably Feb. in your area.)
Is the grass seeding in or coming in from the edges? If it is from the edges, I would stop it now by edging the beds. If you don’t have an edger, a flattish garden spade will give you a pretty clean line too (it is what I use in my yard.)
Grass seeding in is a bit trickier as you may want to wait until more of the perennials are up before mulching since it is a new garden to you.
Don’t bother weeding until you actually have the mulch on site so that you can mulch as you go, and not give new weeds any chance to pop up! Or depending on how bad the weeds are-you may be able to get away with no weeding. Simply cover the space between perennials with several (at least 10) wet sheets of newspaper and then with mulch. This will smother the weeds between the perennials; if you have weeds growing in the perennials it won’t work as well.
In the veg. garden if you planted in rows it will be easy to use the newspaper method there as well. Make sure to use something that will break down quickly to mulch in the vegetable garden, such as mushroom compost or hay.