Nancy,
There are a multitude of small cities & towns in the Province of Sevilla, which you can easily reach via bus, or Cercanias (commuter train) departing from Santa Justa's train station.
We visited Sevilla this past October, & I had considered taking a couple of day trips nearby, unfortunately, since we only had 5 days to explore, we didn't take any day trips. I know, I know... I was traveling with my mom who is in her 80's & she preferred to stay in Sevilla.
Do you speak Spanish? The information given below is found in the Renfe website but it's only in Spanish.
You may want to read about these places reached by Cercanias trains. I'd considered going to Utrera & Lora del Río
I'd suggest you pay a visit to the TI (tourist information) office once you get settled in Sevilla.
Here's another tip : If you will be taking the bus within the city of Sevilla, you may want to get the 'Tarjeta Multiviaje' which will give you discounted bus rides (€0.69 or €0.76) vs. paying for individual tickets each time you ride the bus (€1.40)
If you want delicious authentic tapas, you must go to our favorite place... 'La Antigua Bodeguita ', in Plaza del Salvador across the way from the church by the same name. It's a favorite spot for locals, so go early for lunch or dinner.
Enjoy Sevilla!