I am not a fan of a lot of mulch (perennial bed or other).
You can certainly put SOME mulch to control weeds etc, and compost is always good. True, not around tree.
top soil, garden soil and lawn soil can be pretty much the same depending on where you are and whether you have a lot of clay etc. Potting soil is used for containers and has no value as far as nutrients.

I am a great fan of vegetative mulch, as it helps retain moisture as well as eliminate most weeding, but I would not apply it before plants have grown enough to avoid being covered. Al

Ahhh I forgot to post the fertilizer question.... how long do I need to wait after transplanting new perennials to fertilize?

If planted/growing in healthy soil, with adequate moisture for their needs, most perennials generally don't require fertilizer. My peonies, daylilies, coral bells, hosta, butterfly bush & baptisia, among many others, have never been fertilized. Most have been thriving here 30+ years.

If you are mulching your perennial bed to suppress weeds, I've found that spreading 3 inches or more of bark mulch on top of recycled corrugated cardboard works better than any other method I've tried.

I mark where my perennials are growing so have only once accidentally mulched over them. A few weeks into the growing season back in 2011, I noticed a mulch volcano forming in one of the beds. When I checked it out, my balloon flowers were pushing up the cardboard and the mulch--it never even slowed them down.

I don't mulch up against the stems of either perennials or shrubs.

Shadeyplace answered your soil question.

What perennials do you have?

I would not use bark mulch on perennials as they seem to be bothered by the woody stuff. You can use dry, thinly spread lawn clippings, chopped off perennial pieces...
I always have a thin layer around my perennials which love moisture and good feeding.

I would never mulch Lavender or anything with silvery foliage.

And I would just expect the mulch to keep out annuals weeds spreading by seeds, everything else needs weeding (or re-applying of cardbord)

In my established border I hardly need to weed, just mulch lightly to improve the soil and keep the worms happy, the Echinacea, Asters, Daylilies, Phlox paniculata, Geranium and some cover the soil completely

only draw-back is a constant battle against slugs, I use slug pellets

on newly planted perennial beds I use mulch made of chopped hemp (it is sold as litter for rabbits or hamster cages). I apply a layer of about an inch, and it just helps the young perennials to get started and keeps out seedlings of all kind.

I use shredded bark mulch extensively throughout my ornamental gardens on shrubs, trees, and perennials. Works great, keeps down weeds, and the breakdown improves the soil.

I am also a fan of bark mulch. On all of my perennials. About 1-2" thick. Scratch the surface if the ground with a hand rake before you apply it. You will be amazed at how the soil "eats" the mulch by the end of the season. If you buy a good quality mulch it is fantastic for amending the quality of the soil...particularly here where the soil is naturally a very dense clay. Also my worms love mulch. I think the take it as a personal challenge to eat it all up by the time winter is here.
I usually mulch before Memorial Day. I find that it's best to get a jump on the weeds. My perennials pop up right through the mulch. I have never bothered to mark them. If you use cardboard though it's probably a good idea.