Rulebook states, that food cards boost a specific skill. When I activate my tumble skill, I have the boost (for example +2 from Swiftgrass potion) and can tumble 3 spaces (1 naturally + 2 from potion). Do I have for my second skill also 3 movement or only the natural 1?

Food cards apply only to a skill you TARGET (hence the skill icon in the top right corner of the card). therefore only the skill you assign the “boost” from the card too has additional movement range. Other skills are limited to your normal movement range (how many move icons in your upgrade track). Only if the card specifically says “all skills” do all skills receive the boost.