TransUnion is one of the major credit bureaus in the US. Along with other major credit bureaus like Experian and Equifax, TransUnion provides banks and other lenders information about your history as a borrower. When you miss paying a bill, or when you pay your bills on time, many banks report your payment behavior to a credit bureau like TransUnion, and some even report to all 3 of the major credit bureaus. The bank that issues your credit card also reports to the credit bureaus, and paying your credit card bills on time plays a big part in building your credit history.
FICO on the other hand is not a credit bureau. Unlike a credit bureau which provides potential lenders with your detailed credit history, FICO provides a score based on data from the big credit bureaus that shows lenders how creditworthy you are right now. Although credit bureaus now usually provide a credit score based on your credit history at that bureau, FICO creates a credit score based on your scores from all 3 major credit bureaus, offering lenders a more comprehensive picture of your creditworthiness.
Because your FICO score provides a bigger picture of your credibility as a borrower, this is the score that most banks and other lenders will look at. A TransUnion score will provide a good indicator of what your FICO score may look like, but you should count on your FICO score being a fair bit lower than your TransUnion score. For example, if your TransUnion score is 750 your FICO score might be as low as 700 or even 690.
TransUnion does offer additional perks when you sign up to access your credit score, such as email alerts that inform you of any critical changes to your TransUnion credit and up to $1,000,000 in Identity Theft Insurance. You also get unlimited updates to your TransUnion credit score and monitoring and personalized debt analysis and credit trending. Checking your score won't have a negative effect on your credit. You can sign up to TransUnion Credit Monitoring to enjoy these benefits.
However, I would recommend you base your expectations on your FICO credit score, since it is the most widely used credit score in the US. You can get access to your FICO score for a fee through myFICO.
If you prefer not to pay to keep track of your score, you can also get it by using a credit card that provides free access to your credit score. Cards from Discover and Barclays even let you access your FICO credit score.