It's more that they realise that there is an overall economic benefit to you being here, so they begrudgingly accept it and want to keep an eye (or several eyes) on you because, in the words of a policeman I once spoke to "all crime in Thailand is caused by foreigners"
There's nooooooooooooooooooooooooooooooooooooooooooooooooo way in the world I would take a death van all the way to Nong Khai. Here's what I would do:
Fly to Udon Thani. You can fly there for 500b all in (honestly! just check on google flights). Then take a minivan from the airport to the bridge (200b), cross the bridge (there's a crappy old bus that actually takes you the 1km or so across - 15b), get into Laos - you need a visa on arrival - costs US$30, then there is a bus into the centre of VTE (30b).
If your school has got your paperwork in order then I wouldn't bother with an agency, as it's pretty straightforward to go by yourself
I live in Vientiane, so let me know if you need specific advice. It's a really easy place to go and can be done very cheaply and easily, if you know how ;)
Flying to HCMC from Chiang Mai is quicker, easier and probably actually cheaper. Savakhet requires a very arduous journey, or a flight from CM to Udon and then a fairly arduous journey. Bear in mind that you may need a visa for VN in advance, depending where you are from
' advice and head to Laos (presumably Vientiane). You can easily get visas on arrival, it's easy to get there, and while its hardly disneyland there's probably enough to keep your kids entertained for a night or two.
You might get a 'why u come many time' type questioning though, but just give them straight answers (assuming you're not doing anything dodgy, lolz!) and, as Tod says, have your 20k baht and onward travel in hand