How do I clean the the old 4x4 1950’s pink tile and regrout?

Regrout them, it will make a world of difference. Just be sure to seal them so it is easier to keep clean.

http://homeguides.sfgate.com/regrout-old-tile-25244.html

http://www.idealhome.co.uk/bathroom/bathroom-advice/how-to-grout-wall-tiles-floor-tiles-61688

http://www.diynetwork.com/videos/easy-grout-repair-62181

The make grout cleaners and bleach to clean it, they even make specialty brushes,... we always used an old toothbrush and bleach. https://www.amazon.com/s/ref=nb_sb_ss_i_3_13?url=search-alias%3Daps&field-keywords=grout+cleaner+and+whitener&sprefix=grout+cleaner%2Caps%2C298&crid=1IEQ6Y6DHJ6IO

The grout pen is like a paint and mildewcide... https://www.amazon.com/Grout-Pen-Large-White-Restore/dp/B006MDF3ZQ/ref=sr_1_13?ie=UT

Once you get it white, I would seal the whole thing, tile and grout to keep it clean.... https://www.amazon.com/Aqua-Mix-Sealers-Choice-Gold/dp/B0039ZDMLC/ref=pd_sim_201_6?_encoding=UTF8&pd_rd_i=B0039ZDMLC&pd_rd_r=0XDCFAFY22J5DJD6Y911&pd_rd_w=eLyah&pd_rd_wg=gZPs2&psc=1&refRID=0XDCFAFY22J5DJD6Y911

Grout can be painted. Clean really well with a baking soda and hot water paste. Clean with 1 part vinegar and 1-1/2 part water. Let dry completely. You can use acrylic paint* sold in small containers in art stores. Use sparingly and let dry. Seal with grout sealer. Good luck.

*I advise against white. White is often opaque and hard to completely cover.

You got this! https://www.youtube.com/watch?v=PXRtbRqLN68

Grout is the EASIEST and most inexpensive thing to do!

Regrout! You will be so much happier!