The Cathedral, the Monasterio de las Huelgas and the Cartuja de Miraflores are, IMHO, the top 3 of Burgos' religious architecture. Being pilgrims, you might be interested too in taking a look at the (remains of the) former Hospital del Rey (nowadays is a school of the University of Burgos).
The Monasterio de San Pedro de Cardeña (
www.monasteriosanpedrodecardena.com) could be of your interest too but it's something like 10 kms out of Burgos and I'm not sure how easy/difficult would be to get there for a group like yours.
The tourism web of the city hall of Burgos has a brochure with ideas to spend 72 hours in Burgos. You could take a look at it to get some ideas:
www.aytoburgos.es/archivos/turismo-en-burgos/articulo/documentos/burgos-in-72-hours.pdf
If you insist in visiting a winery, the Ribera del Duero wine area could be of your interest and
www.rutadelvinoriberadelduero.es a good web to get info about it. There's a wine area closer to Burgos: the Arlanza wine area but I don't know too much about it. You could use its web (
www.arlanza.org) to ask for info though.
...and an olla podrida for lunch.