Tile first. A plumber shouldn't have a big problem changing the location of the plumbing lines.

Where are you putting it, and show a picture of the existing bath. please. Even bath freshening needs a good plan, even those with restrictive budgets. A vinyl floor leads me to believe that is what you are attempting. Show the existing conditions.

And since your home is from the era when much of the vinyl flooring contained asbestos, have a small sample of your floor tested by an accredited asbestos test lab; it should cost less than $100. If it tests positive for asbestos, you should have it professionally removed by a licensed asbestos abatement firm, or cover it with a new subfloor before tiling.

Instead of buying a pre-made vanity with a top included, order cabinets and have the top made with the sink to one side. I think it's nice to have it offset a bit so you have room to spread out on one side. I think this may be cheaper than having plumbing moved, too. I had a longer vanity with sink on one side in previous home and I had a cultured marble top made exactly to my specifications. Should be able to do the same with a granite top and ceramic undermount sink if you prefer that - you may even be able to find a granite remnant since you only need 4x2'.

Actually the vinyl floor was probably replaced around 1990. So it's just the vanity that is the same as when the house was built in 1958. The sink has developed a hole in the bottom that cannot be repaired. So I thought it might be time for a whole new vanity. I do, however, have the same green tiles on both the vanity and around my bathtub now, so am wondering if I should also replace the tiles around the tub, so it won't look so "old fashioned".

The tub tiles are in perfect condition, but do not go all the way to the ceiling. The vanity tiles have a couple cracks around the sink border.

If the floor was replaced around 1990 you should not have to worry about asbestos in the floor, unless the old tiles were simply covered over and not actually removed, which is a not uncommon practice. In addition, you should be careful about pulling the existing floor covering and sanding the floor in case there are remains of old adhesive, which also often contains large amounts of asbestos. Sorry to sound negative, but I just wanted you to be aware of what you might be facing so you can make informed decisions.

P.S.: Thanks for the trip down memory lane: that pink and gray combo reminds me of a tie I had in high school that made me look cool many years ago. :-)

kudzu: My other bathroom actually has a vanity with light gray tiles and a pink/peachy color sink.