painting the tiles can make a huge difference. however it really is worth investing a little more time and money to get the best finish. i have found that using tile paint alone meant that the finish chipped quite easily - annoying when the tin says it is a one-coat system.
i would use zinsser BIN as a primer, it's shellac based, you don't need to sand the surface (although you must make sure the tiles are thoroughly clean and dry before you start)it also dries quickly and you can paint over it after 45 minutes. this primer gives a really good, smooth base for the top coat - i would use tile paint as it is in a bathroom - and also improves the adhesion of the top coat making it much less likely to chip. although paint looks and feels dry after a few hours it will not be properly cured for a few weeks so it is well worth being careful for 2-3 weeks until you are sure the paint has cured.
BIN is quite expensive but it goes a long way and gives such a good finish i think it i worth it. especially if you need the tile paint to last a couple of years.
for the floor - it depends what is underneath. if it is wood then you have the option of exposing that if it is in good condition. painted floorboards can also look very good. or you could go for a vinyl again - most bathrooms are small enough to be able to buy an offcut, you might be able to find a nice quality offcut. rubber floor tiles can also looke quite good and they may not be too expensive if the floor space is not too big.
if you have a grey plastic bath panel to match the suite then you could replace this with an mdf panel painted the same colour of the tiles. another small change that makes a huge difference.
i have seen some simple bathroom makeovers where the avacado/peach/yellow/anything but white suite has looked amazingly better once the walls and floors have been neutralised.
i would think white tiles would look best as they would give a crisp contrast with the grey suite. you could then add in any colour accessories and towels you like - brights probably best as anything too pale could look a bit insipid. chuck in a few nice stainless steel accessories and i bet it would look great! if the taps have seen better days then it is amazing the difference that replacing them can make - you can pick them up quite cheaply in the sales and fitting them is a diy job if you are any way handy.
sorry for the very long post - i think i got a bit carried away! good luck with your bathroom makeover, let us know how you get on.