Recs: 
Henri- inexpensive hotel option by the train station, can take the train back to the slope or take the bus. Has a breakfast option. Rooms are large and staff is great.
Les Deux Kitzbuhel- Michelin star , located right in the centre of town 
Tennerhof Hotel- A relais chateau hotel located on the "bunny" mountain this is a bus ride into town. Stunning hotel with great views. An excellent restaurant on site with a cute bar! 
1st lobster- cute little restaurant located in town, casual vibes that serves you guessed it, lobster
Kisini- coffee shop in town, has gluten free options too! 
CENTRO- Italian casual spot in town centre
Huberbrau-Stuberl: austrian cuisine 
Restaurant Bergerricht: michelin star restaurant 
Zuma- if you need a change of food, this is the chain sushi spot! 
Berghaus Tyrol- mountain spot for lunch 
Sonnebuhel- of the top apres spots & lunch spots. Check if they have any special events when visiting. We went to the Valentines special dinner and it was a treat. 
Restaurant Hochkitzbuhel bei Tomschy- mountain lunch spot
SeidlAlm am Hahnenkamm- mountain lunch spot closer to the bottom of the mountain