It's not *acrylic* per se that I avoid, it's cheap yarns. They tend to be stiff and scratchy.
You mention Red Heart, which is a brand that has a number of yarns it its line and has "sister" lines (TLC, Moda Dea). There are several yarns in those lines that I have used and that I like, but Red Heart Super Saver is not one of them. I've found that the cheap worsteds don't work up to suit me, don't feel good while I'm working them, and don't wash out soft enough to suit me. They do wear like iron, though, so if you're, say, crocheting a heavy mat to stand on while you're cooking or washing dishes, Red Heart Super Saver might be a good choice for that. One other caveat: If you leave a skein of Red Heart Super Saver in your stash for a couple of years, it will get harder and harder and harder. Ask me how I know this.
My favorite inexpensive worsted-weight yarn is Plymouth Encore. It comes in a huge array of colors and is widely available at yarn shops or on line. It retails for about $5.50 or less for a 100-gm/200 yd ball. It's 75 percent acrylic and 25 percent wool, so it has many of the good characteristics of both: machine washable and dryable (I dry socks in the dryer but hang sweaters or lay them flat) from the acrylic, warmth and loft from the wool. If you're making a garment, I recommend you give it a try. You can make a child's sweater from 3 balls and determine how you feel about it for an investment of around $15 or a pair of warm "boot" socks for $10.
My absolute favorite yarn to work with is Lopi. However, I live in a part of the U.S. where we do not have enough cold weather to justify using Icelandic yarn. I get my Lopi "fix" by knitting for Warm Woolies and other charity groups that need warm items donated.
Personally, I find buying cheap yarn for everything a false economy, partially because it decreases my pleasure in knitting or crocheting and partially because the life of the finished product is often shorter. But matching the yarn to the purpose will help even that out.
Hope this discussion helped you decide.
Edit: On the dog sweater: without seeing the pattern, it would be difficult to say. However, when you said "dog sweater," I thought of TLC Heathers. Personally, I prefer the colors and the fabric; your mileage may vary.