What do I need to effectively clean my records

I have some new and used records that are in need of cleaning. I'd like to buy some materials I can use from home, and I don't have space or money for a cleaning machine. I took a couple to the local store and cleaned them on the VPI there, didn't seem to do much. A guy there recommended I buy a more effective cleaning solution and stiffer brush, but then another guy said a stiff brush would damage the records. I read the post about the sketchy "spot remover" type cleaner being used, but I'd like something guaranteed not to harm the records or my rig. I keep hearing about Last, both the cleaner and protectant. Is the protectant worth it?

I'm a total novice. I have only a dry brush. Can I get some advice on exactly what to buy and how to clean: brands of cleaners, what to scrub with, what to dry with, how to clean them (use circles? brush inside to outside? how vigorously?). It sounds silly, but where do I set the records to dry? Also, should I buy replacement audiophile sleeves (which ones) to keep my really nice records in? Is a separate stylus cleaner necessary?