OK
WL, will do, makes sense.
It also makes sense to start this assuming anyone this could apply to is starting off this adventure from Hua Hin. Some of this is self-explanatory or obvious but will include it all in case there are some that have never been. Check whether there are any official Thai/Lao holidays around the dates you want to go, the embassy will be closed for all of them and there are quite a few.
1. First thing to consider is, how are you going to get there?
-If you are going to drive, then you go to Bangkok obviously and then head northeast on Mittraparb Rd(A2, I think) which goes through Korat, Khon Kaen, Udon Thani and then on to Nong Khai, don't get off it, goes all the way. Average time lets say HH to Nong Khai(driving), maybe 10-12 hours at an average pace, it's a good 800km or so, give or take a bit, at a guess. Korat would be a good place to spend a night if you want to take it a bit slower, lots of good/reasonably priced hotels there. Korat would probably be about half way. There are places you can leave(in Nong Khai) you car/truck for a night or two which are safe while you go to Lao. Firstly there is a secure car par at the bus station which will charge you one hundred baht or so a night or you can rent a cheap hotel room which has a car par plus security and leave it there, done both myself, no worries with either.
-The bus is also another option but it's a long way and that bus trip would not be for the faint hearted. I'm not a big fan of the bus but some are. I gave up using the bus a while back for many reasons.
-If you're not driving then I'd probably go with the train, at least once you are in Bangkok. There's an overnight train(sleeper/1st class), not too expensive, that leaves Bangkok main train station in the early evening, not sure exactly when it leaves but it's going to be around 8 o'clock as it gets to Korat about 11-12 o'clock/midnight. It gets you to Nong Khai in the early morning about 5-6 am, which is perfect, as the bridge opens at 7:00-7:30 am. If you are hungry and want a recognisable breakfast then you can hop into Nong Khai(40-50 baht tuk-tuk) to the Pantawee Hotel which has a 24 hour restaurant open, food OK.
2. Cross the Friendship Bridge when it opens on the shuttle service, not too busy then. Get the visa to Lao on arrival, don't need the touts, it's easy. The office for a Lao visa is next to where the shuttle drops you off(Lao side of the bridge). Lao visa $31 or 1550 baht, cheaper if you have US dollars.
Taxi from the bridge to Vientiane about 300 baht, any more and they are trying to
lift your skirt up.
3. Good idea to go straight to the embassy and get your application in asap. It can get very busy at the embassy and usually there are lots of people that don't have a pen, can't fill the form in properly, have no idea what kind of visa they want/need etc etc

.
It says that you have to wait two working days for the return of the passport with the visa in it. This is somewhat misleading because if you apply on the morning(as decribed above) of, say, a Wednesday, then you go and pick it up in the afternoon of Thursday, the next day.
You can only apply for a visa from 8:30am(when the embassy opens) up until 12:00 midday, each day, cannot apply in the afternoon. 13:00 -16:30 everyday is used to hand back the passports to the people that applied the previous day.
Do not apply on a Friday unless you are happy to wait until Monday to get it back.
Do not lose the reciept they give you at the embassy when you apply. If you lose it they will make you wait till the end of the handing out session the next day and give you the Spanish Inquisition routine before you get the passport back, definately a massive interjection into your plans.
4. Good idea to download the visa application form from the embassy site and fill it in before you leave and have everything ready and done when you arrive, then it's just hand it over and pay the fee. Quickly do the aformentioned and get the hell out of there to the hotel of your choise. Lots of stuff around the fountain in the center.
Just tell the taxi/tuk-tuk driver you want to go to a restaurant called "Cop chai ders"(can't remember the spelling but that's how it sounds) and that will get you to the right area, the choise of hotel etc after that is yours but plenty there. A good place for breakfast/lunch in this area(next to the fountain square) is a place called the Scandinavian Bakery, famous and decent, nice food.
5. The following day, get to the embassy to pick up your passport at about 11:30-12 midday. It's much quicker to get it back than the process of applying. Then it's just the taxi(3oo baht) trip back to the bridge and hop over to Nong Khai.
However, this is were problems can start, the trip home, unless you are driving yourself. If you look at the train times of trains leaving Nong Khai you'll find that the afternoon train leaves at 13:00 which is no good for you as you have to pick up you passport in Vientianne at that time, missed it. The next one leaves in the very late afternoon/early evening.
Really, you need to get to Udon Thani train or bus station. You have two options. Take the bus from Nong Khai to Udon, which is one of those very slow buses that stops every 2 km to pick up someone and takes an hour to get to Udon and is not guaranteed to take you to the bus station anyway, you'll probably have to then take a tuk-tuk after it stops at the discretion of the driver to get to where you need to go.
The other option is to take a private taxi at the bridge to Udon, more expensive, but gets you straight to where you want to go. Otherwise, stay in Nong Khai for the night or just wait for the next train. You choise but this is just the senario, do as thou wilt.
If you end up in Udon, at the bus or train station then getting back to Bangkok is simple.
These are just a few observations and red flagging a couple of potential problems that may come up on this visa trip.
Couple of tips:
Take small denomination notes of baht or US dollars, it's very useful and you can easily use these currencies everywhere. If you change a 500 baht or 1000 baht note then they may be unable to give you the change in baht, thus requesting you accept kip(Lao currency). You really don't want kip and if you get then use it asap and don't have any left when you get back to Thailand, no bank here will change it, it's worthless.
Check out the duty free shop on the Lao side on the way back to Thailand, it's great, cheap as hell. A 70cl bottle of blue label vodka is about 280-300 baht, bought 4 last time I went,

, limit is two but they never check.
A liter jug of beer Lao will be about 9000-10000 kip/35 baht/$1 US maybe a little more now.
Exchange rate will be about 240 kip to 1 baht. You can pretty much change money anywhere, even restaurants/bars.
The location of the embassy has changed as
WL said but that was about 5 years ago so should not cause any problems now.
Will have probably forgotten some stuff but any specific questions then feel free to ask, will do my best to answer, have done this trip about 20 times now, due to all those tourist visa runs and more recently Non O runs over the past 8 years or so.
Lomuamart does make some good points in his post aswell, well worth paying attention to it.
The above is a helpful read if you decide to go to Vientianne for a visa. Please remember that embassies/consulates do change their MO at will and at short notice. It can also depend on whether the official you are dealing with is just having a bad day doesn't want to be helpful. There are many variables.
If you go, I hope this is of some use.
Edited for clarity.