Cow manure is fine to use. By the time you buy it in a bag, it's usually pretty well composted. If someone were to give you a truckload of fresh manure, you would be better off composting it rather than using it 'raw'. Composting helps sprout and rot the weed seeds, and it also helps break down any medications they were given (most cows are not grown organically).
Weeds are best kept down with a layer of mulch. You can use a hoe when both the weeds and your plants are small, but when the plants get a little larger, cover all the bare soil between them with mulch (straw, hay, even dead weeds). As long as you can block the bare soil from getting sunlight, most weed seeds won't sprout if the mulch is thick enough (work up to 6 or 8" thick as the plants grow). Any weeds that do sprout are easy to pull, so just pull 'em and lay them on top of your mulch to dry out and die, at which time they become mulch.
The mulch will also shade the soil and help to keep it more moist than bare soil, and this prevents stress on the plant from the too wet/too dry conditions. Stress slows plant growth down, and it can be from not enough nutrients, not enough sunlight, too much water, not enough water, weed competition, etc.
Straw and hay varies by source, but some of it contains quite a few seeds. If you happen to have a few chickens, toss the opened bale into their yard and let them work it over. They will have a great time scratching through it and picking out all the seeds, and they will be manuring it a bit, too. Then move it all to the garden.
For perennial plants, you can use straw or hay, but most people use something like bark. If your plants need some nitrogen, get a large bag of alfalfa meal from your local feed store (call around --- not all places carry the meal), scatter it under and around your perennials, then spread your bark mulch on top.