First of all, welcome from Michigan!
Quilt as you go (QAYG) is a blanket term for many different techniques. Some involve hand stitching, many do not. Some use sashing (also called lattice) others don't. Some lend themselves to quilts made from blocks, others from strips. There are many you tube videos on different methods.
The tutorial forum on this board
http://www.quiltingboard.com/sitemap/f-10.html
has several. Here are the most recent:
http://www.quiltingboard.com/tutoria...s-t276464.html
http://www.quiltingboard.com/tutoria...l-t283918.html
http://www.quiltingboard.com/tutoria...h-t285382.html
http://www.quiltingboard.com/tutoria...l-t285474.html
My advice would be to take a beginning quilting class at a quilt shop. Even if you are a confident seamstress, there are a enough things unique to quilting that make a class worthwhile. The class also will go over finishing: how to sandwich, quilt and bind.
As mentioned lap quilts should be do-able on a domestic machine. You can do straight line quilting or gentle curves with a walking foot.
Longarm quilting prices vary by geography. Check with your local quilt shop (LQS) or contact a guild to get the names of quilters in your area. Many have brochures with pricing.
Finally, if you don't take a class, before starting one of your gift quilts, make a small quilted project or two, such as a place mat, to try a couple different techniques.