Orlando's Lynx bus service has a route to get you from the Disney
Resort to SeaWorld in about 30 minutes. From the Walt Disney World
Transportation Center, you must get on a bus #50 toward the Lynx
Transportation Center. It will drop you off near the SeaWorld main
gate. A single fare ride on a Lynx bus is $2, as of publication,
making it the most economical way to get to SeaWorld from the Disney
resort. Make sure you have the exact amount, as bus drivers do not
give change.

mearstransportation.com

By far, the least expensive option is the city bus. The city buses are
called Lynx in Orlando and their website is:
http://www.golynx.com. This website will allow you to check the bus
schedule. Keep in mind, buses make stops, so your traveling time may
vary. Some folks report a 20 minute trip from WDW to Sea World via
Downtown Disney for $2 per person each way. Other folks say the trip
may be more like 30-40 minutes. The bus stop at the TTC is sometimes
slow loading and makes a stop at Downtown Disney along the way. The
bus stop at the Downtown Disney lot is on the west side near the
overflow parking lot. You will want to ask the driver to drop you off
at the front gate to SeaWorld otherwise you might be in for an extra
walk.