I find that primer dries so fast that you have to do the job quickly, but it's worth it in the long run to get it somewhat even: any time saved by doing a poor job of priming will be more than made up for trying to get the finish coats looking good.
The primer coat doesn't have to be perfect, but it should cover the surface (no bare spots) and it shouldn't be so blotchy that you get drips or visible unevenness.
Patching:
When patching smaller holes, a 6" taping knife should suffice.
![enter image description here](https://i.stack.imgur.com/5tZli.jpg)
Apply the compound in a few light coats rather than a single heavy coat, and sand between each coat.
You can either sand the patch using a sanding block like the 3M® Fine/Medium Large Drywall Sanding Sponge,
![enter image description here](https://i.stack.imgur.com/BOkw9.jpg)
or wet sand using a slightly damp rag. If you choose to wet sand, let the area dry before applying the next coat.
For larger holes, you'll have to fill the hole with something solid before you apply compound (don't try to fill the hole with mud). Once the hole is filled, start with a 6" taping knife for the first coat. Next, apply a few thin coats using a 10" taping knife
![enter image description here](https://i.stack.imgur.com/ZehyM.jpg)
feathering the compound out to blend into the wall.
When working with drywall compound it's a good idea to use a Hawk, or a mud pan instead of working directly from the bucket. This will insure that you don't end up with little bits of junk in the compound, and that the compound in the bucket won't dry out too much while you're working. They are also good for catching falling mud.
![enter image description here](https://i.stack.imgur.com/UysCk.jpg)
![enter image description here](https://i.stack.imgur.com/YLl4b.jpg)
Preparing the wall:
Wipe the wall down with a dry rag, to remove any dust and dirt. Prime the whole wall with a proper primer, like Glidden Gripper.
Any time you patch, it's a good idea to prime the entire wall. If you simply prime the patches, you may be able to see the patches when you're done.
Painting:
Apply 2 coats of the desired paint, and you're done.
If the patches were done properly (smooth and blended appropriately), after primer and two coats of paint you'll never know there was a patch there.
NOTE: Products mentioned in this answer are for example only, I do not specifically endorse or recommend the use of any products mentioned.
Best Answer
Could be that your mud wasn't mixed well and had settled a bit, leaving excess water on top. A firmer application or a more thorough mix might've helped prevent bubbles.
At any rate, just skim it again, and press firmly using the edge of the blade. If you do it right it won't even need sanding (or maybe just a wipe).