In the old days your had to cross the mighty Mekong river by boat from the immigration office at Mukdahan. The immigration office is still there and the boats are still going but now only for Thai and Lao people (and a lot of cargo). Thai and Lao can use a border-pass (I think it costs 20 thb). Ferry costs 50 thb.
Now there is a bride (5 km north of Mukdahan) which can be crossed by car or bus (no motorbikes!) You can take the 'Savan Vegas" bus shuttle service from the bus-station at Mukdahan (all the way to the casino or just take a TukTuk to the bridge and cross immigration and take the border shuttle-bus (50thb). And there is an bus going from the bus-station in Mukdahan to the bus-station in Savanakhet. The bus costs 50 thb pp/one-way.
The process is simple. From Thailand you walk on the left side and buy a bus-ticket (50thb) then you proceed to exit Thailand via the immigration. Then you cue up for the shuttle bus to the Lao side of the bridge. At the Lao immigration you can get (=pay) a Visa On Arrival (VOA) and then go thru the Lao immigration. After immigration there are TukTuks and taxi's to bring you anywhere. You are then about 2km from Savanakhet city.
The way back is about the same. Be aware that when entering Thailand over land without a visa you are only entitled to a 15-days visa-free entry-stamp. You can cross the bridge with your own private car, but you will need some papers as the car-registration-book, owner of car has to be with the car (or a letter of permission) and you will need an international transport letter. Officially you will need an international driver license in Laos but it seems that a Thai driver license is also accepted.
Savanakhet also has an airport (ZVK, just south of the city) with flights from Bangkok and Vientiane by Lao Airlines.