I have created a logo in Illustrator.
In which file format / document settings should I save it so I can use it for creating a card in InDesign (print) and for placing it in a website?
For saving the file always use native file formats like EPS for Illustrator. That's for your file back up.
For print it is important that you use a
CMYK color mode. Now-a-days
EPS are used for printing. PDF is increasingly used now among designers. TIFF is used with ease since color management is possible but ask your printer if they accept TIFF.
For web logos use
For print I'm sure the other answers here have it covered, but personally i would ask your printer first.
For the web ... the web is starting to go SVG recently - you should ask the web developer - they may want to use SVG because there are lots of possibilities including support for responsive logos, and they're quicker to load - but normally/traditionally it's PNG, and even if you supply an SVG to a developer they'll also need a PNG version for fallback
you already won %99.9999999999 of the battle by designing it in Illustrator rather than Photoshop. As long as you have a VECTOR file of the logo you'll always be able to export it into whatever format you need. Every medium requires its' own type of format, color profile and or size to produce a good quality result so there's more than just 1 or 2 useful formats to export. As long as you have a vector version, you'll always be able to produce a top quality file of any format, so just take it as it comes. As far as indesign, you can just place the .AI file and it will render when you export the Indesign file into whatever format you need it to be.