The first step to making that look better is hand cleaning - hands-and knees on the floor with a bucket of water. I use liquid Mr. Clean in water and microfiber cloths.

From there, I would vacuum the floors before mopping, including cleaning the edges with the round brush attachment on your vacuum. This picks up the loose dirt that otherwise gets pushed into the corners. Vacuuming can sometimes take the place of mopping.

You might want to consider giving up the mop and bucket altogether and switching to steam cleaning your floors. There are several posts here on steam cleaning. It's soooo much easier than mopping.

Cleaning it, as previously suggested, with be a hands and knees project. You might want to think about using an old toothbrush as well. Another tool that might be useful is something the car detailers use. It's essentially a round paint brush but the bristles have been cut back to 3/4" or so. Still flexible enough to get into nooks and crannies but stiff enough to pull out dirt. While you can buy them, you can make the same thing out of inexpensive paint brush and trim the bristles a sharp razor blade As also suggested, vacuum well before you mop.

Once you've got things clean, you might want to think about a nice bead of caulk to create a smoother seam.

What Mike said. Get it nice and clean, let it dry,then apply some caulk.

like graywings mentioned, steam clean.

i have a dry vapor steam canister. its kind of bulky, but great for deep cleaning. i pull it out a few times a year and deep clean walls, floors, everything.

just have to make sure to keep it moving or else it will vaporize the paint right off! wipe with shop towels and that's it. clean.

Yeah, that's gonna take a lot of scrubbing and toothbrushes. First get out all of the debris that you can. I use a steam cleaner and would never go back to a bucket and mop! But, before I steam clean, I vacuum and if I'm having a "good back day" I'll go over it a microfiber dust mop after I vacuum. The floor is basically clean of all debris and dust at this point. I'd hate to tell you how long it's been since I've used my steam cleaner. With my back problem vacuuming and spot cleaning as I go is having to suffice. But, when I do, steam cleaning makes my porcelain tile feel SO wonderfully velvety smooth that I HAVE to take my shoes off for awhile just to walk on my clean floors (lol) Bissell makes a GREAT little stick vac. I think it's under $20 at Walmart and around Christmas time, they can be as low as $14. I have one at each end of the house.
If you decide to look into steam mops, again, Bissell makes a great product! Whichever route you go, always ALWAYS vacuum first! My kitchen tile is 8.5 years old. Baseboards are only 3 years old. I haven't vacuumed in about 4 days. If you get really industrious, you might even consider a fresh coat of paint on the trim. With the clean edges/corners and fresh paint, it'll look brand new :)
Bissell Stick Vac
http://www.walmart.com/catalog/product.do?product_id=14320844&findingMethod=rr#Item+Description

Bissell Steam Mop
http://www.walmart.com/ip/Bissell-Steam-Mop-Hard-Floor/6497961
Monica

Wow. A good cleaning, painters caulk and new paint are gonna be your best bet. Paint your trim with a good quality enamel for the best cleaning potential.