We had the tub in my bathroom painted to change the color from almond to white, when we remodeled my bathroom.. This was done by a professional guy with a sprayer and a paint made for this purpose. You can't tell it was painted......it looks like a brand new white tub. I asked the guy what to use to clean it, because I was concerned about creating scratches or messing it up. He told me absolutely no abrasive cleansers like Comet or BonAmi. He said any liquid cleaner would be ok. My tub has those itty bitty bumps too.......in strips, to help keep you from slipping in there I guess. I've found that Pinesol really cleans it quick and leaves it shiny. It seems to remove the soap scum and whatever without a whole lot of effort on my part. I've also used those Clorox cleaning wipes with good results (was afraid to use straight bleach because I worried that it would eat right thru the paint). It's been over a year now and the tub still looks good as new.
Hmmmm I may try the lemon juice one of these days to see how it does.......then my bathroom can smell like lemons instead of a pine forest...........

P.S. I picked up one of those battery operated Sonic Scrubbers on a clearance rack and let me tell you -- that thing really does a number on cleaning grout. The stall shower in my husband's tiny bathroom never really dries out and the grout gets icky quick. This thing makes cleaning it a heck of a lot easier. I've used it on the grout on my kitchen tile floor too and was amazed at the amount of dirt that came up and how much better it looked afterwards! It would probably work well on those gritty parts of the tub, too.