Grouting travertine generally means filling not just the spaces between the tiles, but also the pockmarks and holes on the tile surface. These naturally occurring holes are part of travertine's unique look, but they also will gather dirt and moisture if left open, so they're usually grouted (with darker grout, so the patterns are still visible). The stone has to be sealed first so the grout doesn't stain the surface, then sealed at the end to protect the grout.
Clean the travertine with a well-wrung sponge, to take up any dust or surface grime. Let it dry for 24 hours.
Apply sealer with a finishing paintbrush. Brush it on in a thin, even layer. Don't get it in the pockmarks and holes on the tile face, or in the spaces between them. Let it dry for six hours.
Use a grout squeegee to spread grout over the travertine. Scrape it over the surface while pressing it into the holes and lines. Let the edge of the squeegee flatten it all out.
Allow the grout to sit for 10 minutes. Use a damp sponge to wipe up the surface, taking the residual grout off the face of the tiles without digging into the holes or spaces.
Allow the grout to set for five days. Brush on two more coats of tile sealer, putting it over the tile and grout both. Let the first coat dry six hours before applying the second coat.