"Mulch" is anything applied to the top of the soil for various purposes, weed suppression being only one. Many folks mulch with whatever they have at hand - grass clippings, shredded leaves, wood chips from the local utility/tree trimming crew, etc. And there's no reason you can't save yourself a step and mulch with compost, yours if you have it or purchased if you don't.

I only use compost as a mulch throughout my garden - adds to soil texture and fertility (never need to fertilize), conserves moisture, insulates against temperature swings and suppresses weeds. And it looks good. It does breakdown or shrink faster than some other mulches but I don't consider that to be a significant disadvantage. I typically spread the compost twice a year (spring and fall) but you could get by with only once.

Less than 2 inches of mulch is a waste of time and more than 4 inches is excessive. And look into bulk soil providers - purchasing compost or any other type of mulch or soil amendment in bags is very expensive. Bulk purchases will run a fraction of the cost of bagged.

It's always a good time to add more mulch or compost :-)

If your beds aren't that large then maybe buying compost by the bag is ok. If you think you need at least a cubic yard then bulk purchases is probably better especially if you have access to a pickup truck that you can use. If using a bulk provider and you need it delivered you'll need to see what the delivery fee.

Whenever buying bulk compost I like to go to the yard and see, smell and touch it first as well as asking a few questions upfront about how it was made.

I typically but the compost at the nurseries which is-I believe- cottonwood compost. I like the look of mulch around the yard so should I rake back mulch, put compost around plants then put the wood compost on top? I don't think the wood mulch is as fertilizing as the compost is. At some stores they sell cow manure. Is this just as good as the bags that state organic compost?

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You can mulch compost on top of existing mulch. Mulch on top of that if you like.

Mulches break down and merge into the dirt. Compost quickest of all, then grass clippings, then leaf mulch and wood mulch takes the longest. All those mulches eventually merge into the soil and feed it.

When it starts looking like dirt, mulch it some more.

Feed the soil and it will feed your plants. It just ain't miracle gro which is no miracle and no grow beyond the month you apply it. Give it time and it will pay off.
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Mulches, of vegetative waste (shredded leaves, wood chips, compost) are digested by the soil bacteria and added to your soil and that becomes the humus, the residual organic matter in the soil. Just put your compost on top of what you use for mulch and cover that with more of the mulch, if desired. Most soils need more organic matter, a desireable level of humus is 5 to 8 percent, so how much to add depends on what is there. Many of us that have been adding organic matter to our soil for years find that all we need to add each year is maybe 1/2 inch of compost covered with much more mulch, the shredded leaves, wood chips, etc.

I think a good time to mulch is in late winter, early spring when you can weed the winter weeds and apply compost around the root zones of plants, then mulch over that. it is easy to see around and work around the plants before they really leaf out.