Question
Best seed for sun and shade?Low traffic.Fertilize when?

Answer
Warm Season Grass of choice for Shade is usually St Augustine -- but low Traffic only.  For Sun, Bermudagrass does best.

Warm Season Grass of choice for higher Traffic is almost always Bermudagrass.  But it needs blazing Sun to do that.

Cool Season Shade Lawns are made of silken Fine Fescue.  It is however the most delicate of Lawns.

For Cool Season Sun sites, grow that old home favorite, Kentucky Bluegrass, which will take a moderate amount of foot traffic.  All are narrowed further to varieties selected for performance at the National Turfgrass Evaluation Program testing site nearest to you.  Wish I knew where that was.

Cool Season Grass of choice for Traffic is thick, tap-rooted Tall Fescue, which is also a champion when it comes to drought tolerance.

Re your Fertilizer question, I must inform you that using chemical fertilizer is old fashioned and outdated Lawn Care.  Traditionally, Nitrogen powder was applied to Warm Season Grass in the Summer, when it was most active and supposedly needed more chlorophyll.  It went on top of Cool Season Grass in the Spring, theoretically to 'green up' the Grass and get it going after a long dormant Winter.  But unless you are trying to keep up a putting Green or golf course (in which case you would have even better methods at your disposal, since these guys are the people who test all the latest theories about chemical-free landscaping), you should instead be building up your Soil long term.

Grass does not need to be 'Summerized' any more than it needs to be 'Winterized'.  Although it is technically a monoculture, a Lawn will be nourished 24/7 by remote control from the ground up.  Sorry, but this is basic science, and easy to do.  Let me know if you would like me to expand on it; I'm happy to do that.

Bottom line: No fertilizer.  Not even the concentrated 'slow release' kind (just a gimmick).

Much more effective: Topdressing, using your choice of Compost or Humus, or one of the items on a short list of nutrients boosters; then topdress periodically with other so-called 'organic' products.  Otherwise, ixNay on the Nitrogen ertilizerFay.

Oh... Ultimate Cool Season Shade Grass is currently (oddly enough) a Kentucky Bluegrass variety, 'Supina'.  It out-grows them all.  Brought to market just recently, therefore pricey and hard to find.  A gorgeous Green in real Shade.

Thanks for your question.  Let me know about that Fertilizer thing when you're ready.  Sorry to have kept you waiting so long; I am backed up something awful here.  Hope this was helpful.