This is the OPO airport:
This is what you'll see after disembarking (if you travel in a low cost company you might get out through a different gate, but the airport is not that big and you'll easily find this area if you have to). Notice, on the right-hand side of the photo, several car-renting companies:
Exit the airport and take the Aerobus, which will drop you about 20 meters from the Oporto train station. The Aerobus hadn't arrived yet when I took the first picture: (note that there's more than one bus stop; you want the one that says "Aerobus")
The Aerobus leaves the airport every 30 or 60 minutes, depending on the time of the day; it does this 23 times overall.
If you want to take a cab instead, here they are, right next to the aerobus:
After riding the Aerobus, you arrive to one of the Oporto train stations; the regular trains are called "suburban trains"; they take around 45 minutes to get to Braga, and they work from 06:15 to 22:30 (24 trains overall). Ticket costs 1,75 euros and you can find the train schedules here. There are also faster trains.
Note that some trains might require you to change trains in the station of Nine (from there there are two different lines to the north, of which one of them goes to Braga). If you have to change trains, the person selling you the tickets should tell you that, as should the person who checks your ticket on the train, but you can also ask (it doesn't hurt and it's safer). Also, your ticket will probably mention the name of the Nine train station.
Please note that the name of the station is "Nine", as you can see by these two photographs (which unfortunately do not include the train station building; only some trains and the lines). It is not the 9th stop or anything! It is just the name of the station! It's a Portuguese word, not an English one! Make sure you get out of the train at the right station, please :) Re-read this line just to make sure you understood it :)
The train station in Braga (which is the last stop of the line, so you're unlikely to miss it, unless the train starts heading backwards) is located near the centre of the city, as is the terminal exchange. From both of those places you can take a taxi to anywhere you want to in the city. A taxi from the train station to the university area, for instance, should cost around 3,50 euros.