There are regular trains going to Santa Cruz from Ferroviaria Oriental. There are 4 services of varying comfort and price. The Ferrobus has Cama (Bs. 250) and Semi-Cama (Bs. 222) which runs overnight leaving at 18:30 pm on Tuesday, Thursday, and Sunday. The Expresso Oriental costs Bs. 127 and leaves at 4 pm on Monday, Wednesday and Friday. The local service "regional" is not advisable due to longer travel times and greater discomfort. It leaves daily at 11:45 am. The Mixto leaves on Thursdays and costs Bs. 47. The station is quite a ways south of the city center.
Note the station of Puerto Quijarro is closer to the border, if in transit to Brazil.
There are buses going all the way to Rio de Janeiro in Brazil, 28 hours away. Crossing the border to the Brazilian tourist town of Corumbá is easy by taxi. Be sure to note border opening hours, the Brazilian border may close early on Sunday.
Taxis will take you to the Brazilian border, from there a bus can carry you to the bus station in Corumbá.
The Pantanal is the largest wetland, and though only 10% of it is in Bolivia, it is possible to see the abundance of wildlife without crossing the border. Bolivian companies such as Ruta Verde can organize trips in advance, though most trips leave from nearby Corumbá in Brazil.
The caves at Motacusito are 7 km to the west, and have stalactite formations.
There are numerous designated swimming holes "balnearios" and waterfalls in the area to enjoy the abundance of fresh water. El Salao and Vertiente, both east of town, are well developed and attract crowds. La Represa, towards the mine of Mutún, is reportedly the best, though at 25kms is far. Do not be fooled into paying taxi fare all the way to the abandoned mine site at Mutún.
Puerto Suarez is the best place to buy the semi-precious gem Bolivianite, especially in uncut form. A large local business has formed around processing, selling and exporting the gem.
There are several restaurants along Avenida Bolívar. This website  has usable information for restaurants and lodging.
Puerto Suarez is not a major tourist center, in comparison to Corumbá, or other Bolivian cities like El Alto. Nightlife is limited by the small size of the city, though Avenida Bolívar is the center of life in the town.