Easy to travel to from Budapest by train or by a mini bus which can be found at the Nepliget bus station. Mini bus is the cheapest option and may be quicker too. Only remember to set your clock ahead by an hour when you cross into Romania.
If you want to visit Alba Iulia you can come using different transportation methods: by train, or by bus. I would recommend coming to Alba Iulia by bus. One of the best company's is DACOS ( http://www.dacos.com.ro ).
Buses come to Alba Iulia mainly from Sibiu, Bucuresti, Timisoara or Cluj. They are comfy, cost as much as an accelerate ticket and get you much faster here. Try to find one from Dacos or Transmixt (part of the Atlassib company) I only had good experiences with both of them.
Buses around Alba - take route number 3 or 4 if you want to ride around the city from the train station.
The railway is maybe the best way to get to Alba for a tourist. The tickets are quite cheap, the connections are very good and from the train station it's easy to take a bus , a maxi taxi or a taxi to see the town.
If you're sitting in a bus station and instead of a bus a mini-van approaches the station , have no fear for these are the good old maxi taxis. They have the same routes and practice the same prices as the normal buses. You can buy a ticket from the driver.
Fortunately, this town doesn't have the predator taxi drivers that you can see in the larger cities- the price is written usually on the door of the car, and I've never met yet one that would try to pretend more money from me than the ride was worth.