I just arrived in Budapest and found that my flat has no Internet. I'm in a long-distance relationship so I need this service to be able to communicate with my boyfriend. What service providers are there in Budapest and do they provide Dongles (the mobile broadband sticks)? Also is the Internet speed that comes with these sticks fast enough to get a good connection with Skype?
Yes, there are plenty of service providers in Budapest that provide the mobile broadband dongles you're looking for - Telenor, Vodafone, T-mobile (to name a few). In the meantime. there's actually plenty of WiFi hotspots around the city that even offer free access if you buy a coffee or a meal. The big malls, like Mammut or MOM Park, are also WiFi zones - just look for the sign.
The connection with the mobile broadband is probably fast enough for a Skype conversation, but not a video call. That said, it can be finnicky, and sometimes the connection is delayed.
Most of the mobile broadband in Budapest comes with a limited number of megabytes or gigabytes. You won't need that much if you're just using it to surf the web and talk via Skype, but downloading music/movies and using video chat can eat up some serious broadband.