I am looking for inexpensive cabinets for my kitchen that won't use up my remodeling budget. Do you recommend assembled or ready-to-assemble?
Rest assured that there's a wide variety of quality kitchen cabinets, both preassembled and ready-to-assemble, available at reasonable prices. They're generally available from one of two sources, and the source you go with will determine whether or not they're assembled for you:
A big box store like Home Depot stocks quality preassembled cabinets in common sizes. Recently I helped remodel the staff kitchen in a house owned by my local Boy Scout council. We were able to do a complete kitchen remodel using Home Depot kitchen cabinets and premade countertops for less than $2000. And this was a fairly large kitchen. There were 8-foot and 10-foot walls of cabinets. My teenaged son and I completed the entire project in two days.
The other quality option, as you mentioned, is ready-to-assemble cabinets. I've had good success with cabinets made by IKEA. I used IKEA cabinets for a kitchen remodel in our office, and found them to be very well-made and durable.
So either way, you can create an affordable kitchen remodel. If you're still on the fence, here's something to think about: Ikea has a lot more design options than Home Depot, and also gives you more custom interior options like pull-out drawers.
Good luck, and show us before and after photos when you're done!