After you visit the Riverwalk and The Alamo, go to the McNay Art Museum, admission is free on Thursdays afternoon, and on the first Sunday of the month. Visit the farmer's market at The Park at PearlSaturday / Sunday and the Witte Museum on a Tuesday is free from 3 to 8 p.m.

Julian's answer is spot on.  The The San Antonio River Walk is fun to wander and stop for a margarita or a bite to eat on the many sidewalk cafes and restaurants.  You aren't far from the The Alamo at all from here.  There are four other missions in San Antonio that you can visit at the San Antonio Missions National Historic Park.  The Alamo Beer Hall has free tours, and they are in the riverwalk area as well.

check out the McNay Art Museum if you have your student id still, its free. Also, walking down the riverwalk at night, although most of those bars are expensive. a great affordable hip place is moses roses on the weekend! if you're going and its the first friday of the month from downtown to south town (its super artsy and fun)..you NEED to try san antonio pub run(just google it). Its probably the most happenin night all month and there's always a theme. the japanese tea garden is free and beautiful. Rent a bike at the Pearl Stable and ride to King William District CrossFit, its a great way to spend the day. make sure to check groupon for cool affordable activites to do downtown or around san antonio.

The San Antonio River Walk, Alamo Plaza, San Antonio Missions National Historical Park Visitor Center, Japanese Tea Gardens, Botanical Gardens, Pearl Brewery, La Cantera Parkway and you cannot miss eating and drinking at the Market Square Lot, Mi Tierra Cafe y Panaderia!!!

The San Antonio Marriott Riverwalk downtown.  It is free and great restaurants. I do not if The Alamo is free???? McNay Art Museum.  I do not know if it is free. I guess I do not now that much.