This a bus/train ride around the city center. They will take you to relevant places in the city while telling you the history of each of them.
Callejon del Diamante is a narrow, steeply inclined pedestrian alley lined with vendors and small shops located one block from the Zocalo of Xalapa. You can find all sorts of quirky and traditional handcrafts, as well as coffee and other things.
This museum has a lot of pieces from the ancient civilizations that existed in Mexico. You can see some of the famous Olmec heads as well as a lot of other art forms from the time.
The Hacienda del Lencero is a Spanish Colonial architecture era building and gardens located near Xalapa. The house, which takes its name from Juan Lencero, a soldier of Hernán Cortés, was the property of Antonio López de Santa Anna in the 19th century. Today, it is a historic house museum which displays furniture and items dating from this period. It also has a chapel and spacious gardens surrounding the property which include a sculpture by Gabriela Mistral who spent time there while in exile.
Coatepec is considered a "Pueblo Mágico" (Magical town in english), this is a accolade given to certain towns in Mexico that are considered to offer visitors a "magical" experience – by reason of their natural beauty, cultural richness, traditions, folklore, historical relevance, cuisine, art crafts and great hospitality. It is only 17 km (∼10 mi) from the recommended hotel and it would make for a great day trip before or after the wedding. The town is very colorful, and has great food and coffee.
About 75 minutes away from xalapa, and very close to the airport is Veracruz City, if you would like to enjoy a day at the beach before or after the wedding, this could be a good idea.