First and foremost, since the culture is not in the greatest health and is a fairly low cell count, proper sanitation is an absolute necessity - even more so than usual.
For the initial step, I like to do it in the bottle instead of pouring the dregs off. Transfers are the times where you run the biggest risk of contamination, so I like to make sure the culture is woken up a bit before transferring out. Sanitize the bottle and bottle opener before opening. Then sanitize the neck/lip of the bottle before pouring the beer. I like to leave about 1/2 inch of beer in the bottle, plus the dregs. this way you get any yeast that is still in suspension and not just the flocced out dregs.
I then use a sanitized funnel to add about 1/2-1 inch of 1.030ish wort. Once diluted with the remaining beer, this gives you a nice low OG of about 1.020. This is less stressful to the yeast than the typical 1.040ish starter wort we typically use. Then I cover with foil (for non-sours) or add a small stopper and airlock (for sours). I usually give the first step about 7-10 days to give the yeast plenty of time to wake up and do their thing.
From there, the general rule for stepping up a starter is a tenfold increase each step. So step two is maybe 200 mL or so of 1.035 wort, and then that can go into a normal 2-liter starter. Use your nose to tell you whether there are any problems, and taste your larger starters to ensure that you didn't pick up any contamination along the way.
For sanitation, I have had good luck simply using Star-San, but if you really wanted to take stronger precautions, then using something like 151 or Everclear, then flaming it off is the way to go.