If it is just one or two of you, then your best bet is Mears Transportation.
From your Disney room, the evening before you want to go to Univerasal, dial "0" from your room.
Have the Disney operator transfer you to Mears.
When the Mears operator picks up, tell then you want to go to Universal the next day, about what time you want to leave, and where you are staying.
They will give you a pickup time.
Be out front of your resort about 15 minutes before that pickup time.
When the Mears bus/van arrives, you will pay the driver in cash $16 - $17 (havn't used this in a couple of years, so not sure of the price) per person. You will then be transported to Universal. It is possible that the van/bus will make additional stops.
Once you arrive at Universal, you may schedule your pickup at that time OR you may call the number on your return voucher 2 hours before you wish to be picked up.
At your pickup time return to where you were dropped off and you will be returned to your resort. It is possible that you will make additional stops before arriving at your resort.
If you would like to pay for this service with a credit card, you can stop at the Mears counter at Orlando Airport and purchase the roundtrip vouchers there. You will still need to call the night before and reserve the bus time.
If there are three or more of you, then a 1-day car rental from
Alamo/National at the Dolphin or Car Care Center will be better price wise.
Also, many towncar services provide transfers to Universal from Disney resorts.