You can take a public bus from Mochit Bus Terminal. I’m not sure how much it costs, but I think it should be around 700THB for a 1st class bus, 350 for a 2nd class bus. You will need to have a visa for Laos to take the bus to Vietiane.
If you do not have a visa for Loas you can take the bus to Nong Khai then get your visa at the boarder. That’s what I did. That bus ride (to Nong Khai) is 10 hours long. So, the bus to Vietiane would be longer that that.
Just wanted to say this is a brilliant and informative site. I am living in Korea and rarely travel away from the block I live in. I plan to visit some of the places you highlight. I have added your link to my blog. Thanks.