How do I batch sparge?

While equalizing the 2 runoffs does increase your efficiency slightly, if they're within even a gal. of each other it's close enough. Also, as you learn, you can use the empirical method. Mash with whatever liquor/grist ratio you like. After you drain the mash, measure how much you have in your kettle. Subtract that from the amount you want to boil. The answer you get is how much sparge water to use. For more details, see my Cheap and Easy Batch Sparging site or my article in Sept./Oct. issue of Zymurgy. Batch sparging is so easy that when I teach classes on it, the usual reaction is "Is that all there is?"

My previous brewing attempt was my first switch to all-grain brewing, so I was right in your shoes. It turned out great, and was a lot of fun. I made my cooler out of supplies from Home Depot using this video: How to make a Mash Tun from a cooler

Then I watched this video: Easy All Grain Brewing - Batch Sparge Method

Then I compiled these instructions for myself:

Mashing

1. Heat water to 168F. One quart of water per pound of grain.
2. Add strike water (initial water) to mash tun.
3. Add malted grains and adjuncts and stir them into water thoroughly.
4. Make sure mash temperature stabilizes at about 155F-158F.
5. Close cooler and let it sit for an hour (possibly throw blankets over cooler to keep heat in).
6. Add hose to mash tun valve and let wort run out into a pitcher. Recycle wort back into mash tun until wort is less dark and grainy.
7. Allow all the wort in the mash tun to run into the boil kettle.

Sparging

1. Estimate how many gallons ended up in the boil pot. Gather enough 170F water to fill the boil pot to 6 gallons (estimate: 4-5 gallons of sparge water for 10-14 pounds of grain).
2. Pour the 170F water over the grains in your mash tun evenly, let the wort flow into the boil pot.
3. Allow the boil pot to fill up to 6 gallons. Do not use too much sparge water!

Another tip is don't use too much grain in compensation for it being your first time. I ended up with way too high of gravity because I was worried about conversion. There's a decent chance you will get good conversion on the first time. Good luck!

I am a batch sparger and I love it.

To address your specific questions. You do not need to stir up the grain prior to vorlauf. Just start drawing liquid out the bottom until it seems to run pretty clear. (And clear means just the large chucks). After that run all the liquid out of the tun. I add more water and stir it up. The stirring is vital at this point to get good exchange of sugar held in the grains into the water as efficiently as possible. There is no reason to let it sit longer, I start to vorlauf right away until clear again.

As for volumes, I mash at my desired ratio. Then I measure how much I collect from the first runnings. I subtract that from what I want my total boil volume to be and add that volume back into the tun. I know there is a rule of thumb out there about trying to equalize the volume of each running, but I haven't found it that important. (And neither does Denny Conn, the dude who has really pushed batch sparging into the mainstream). For example, if I was planning on getting a 7 gallon wort for my boil, and I draw 3 gallons for my first runnings, I know to add 4 gallons for the second runnings. Its much easier this way than anticipating what the first volume is, adding water prior to run off to get the same volume for your second run.

Here is the link to Denny Conn's Batch Sparging website for reference, but note the changes I mentioned about equalizing the two run offs, isn't his recommended practice anymore.