There are 8 different types of buildings that you can upgrade in the game.
Your stadium is where your home matches will be played. As your fan base grow, you need to make sure that your stadium can accommodate all fans who want to attend your matches. Your match revenue from home matches greatly depend on how many people attend those matches, so give priority to building a stdium and upgrading it to a decent capacity before you start building other facilities.
Fan shops are outlets in your stadium to sell club merchandise. Spectators who attend your home matches are likely to buy club merchandise, thereby generating a nice additional revenue. The higher the level of your fan shop, the higher the revenue you generate from it. But keep in mind, match attendance depends on your stadium capacity, so without a decent stadium, your fan shop is not worth what it costs to build or maintain.
The training centre is where your players carry out their routine training activities. Better training centre will have better equipments and training facilities, so your players improve their attributes faster during training.
The youth academy is where the club finds, recruits and develop young talent. Young prospects develop their basic footballing skills at the youth academy, before they are ready to join your club and become professionals.
The medical centre is where you take care of your players' health and fitness. A higher medical centre will keep the fitness levels of your players at a better condition.
Media office is your club's medium to facilitate public relations and deal with media agencies. Your media office is in charge of advertising and live television coverages of your home matches. A higher media office will bring you more advertising and TV revenue from matches played at your stadium.