Call a tub refinisher.

I've linked to the first of three long threads about painting over red cabinets from the Home Decorating forum. She pretty much did what Christophern suggests. She used Ben Moore Advance. Here is a link that might be useful: SunnyCottage's kitchen cabinet painting

Not knowing anything more about your cabinets than you have described, i.e., a 1" recess, I would probably do as you described with a filler. I would feel comfortable screwing into a solid wood face frame, less so into a 1/2" particle board side panel. Use flat-head screws, drill a pilot hole and countersink and they should be unnoticeable from either top or bottom.

Fastest and easiest? Hire a great pro. If not that, I'm not sure there's a fast and easy option unless you're super disciplined about decision making. Here's what I did, roughly, and my tile guy and salesperson said I was faster than most, and the bathrooms look good. Not perfect. But we're happy with them. Since you've selected fixtures and vanities you like, prioritize your considerations. What matters most to you? Aesthetics? Function? Maintenance? Budget? Make a list. Ruthlessly eliminate products that don't fit your criteria. Narrow color and material choices to three for tile, three for counters. Go look. Limit time in stores/showrooms to no more than half an hour (okay, maybe an hour if you've got multiple rooms) at a time. Set a timer. Bring home samples. They probably won't be the ones you thought you'd like. Repeat this process up to three times until you get something that looks good. If you can't find anything you like, hire the pro. Remember: There is no One True Choice. Suspecting there is leads to trials and tribulation. Do not take this path. We're doing four new bathrooms in a massive remodel. The way this process worked in the master bath my husband will use went like this. For the countertops, maintenance and function are most important to us. He's a bad splasher and a bad cleaner. Never sees water spots or that blob of shaving cream he spilled. So counters can't etch or stain. Quartz fits the bill and our budget. He wants white. Not my first color choice, but easy enough. Same will work in guest bathroom and laundry room, thus making efficient use of slabs. Pick the most reasonably priced quality product in the store giving me the nice discount. Be thorough with research, but don't over analyze. Set a deadline for your decision. Stick to it. We chose tile at the same time. Budget was important here, as well as maintenance. DH said he wanted something that looked like a "Roman" bathroom. I interpreted this as marble. No. Not going there with his spilling habit, aversion to wiping things down and cleaning in general. So porcelain or ceramic tile. Fewer grout lines=fewer cleaning woes. I told all this to my salesperson and she immediately directed me to an inexpensive fake carrarra-looking ceramic 18"x18" tile. Works for shower walls. We were going to use it for the floor, too, but a last minute substitution floor tile in another bathroom allowed us to use a grey stone-looking porcelain 12"x24" instead. The contrast looks nice. And we sold the extra 18"x18" to a friend. I started out loving Ann Sacks tiles and New Ravenna mosaics the most, but really, I love tile. The colors, the textures, the possibilities! I could look for hours and imagine all sorts of looks. Then, like everyone else, I had to make real life choices I could live with and afford in the midst of selection fatigue. My salesperson at the tile store was a huge help in narrowing contenders. I found her by looking at Yelp reviews, then went to meet her. She listened well and she knew her products. Great combination, and so helpful. Sorry for the dissertation, when you asked for a nugget. Good luck on your selections. Are you going to post them for us to look at?