Personally I started a couple of days after the milk came in - I think around day 6. They say wait until your milk is a bit more established but I was keen to try and get DD used to the bottle asap. I aimed to express for one bottle a day but had to play around to figure out the best time of day to express to get the 'best' both quantity and quality wise.
Also I got very very little with expressing for the first few weeks, and I had no idea if she was getting the same herself directly, which was quite distressing as there was no way what I could express in one session would be enough for one feed at first. I was advised that the expressing wasn't nearly as effective as her suckling directly so not to worry. This seems to be the case for us (judged by her finishing on the boob and not needing another feed for 2-3 hours).
After a couple of weeks I was also getting 100ml from one side and still only 10ml from the other, (it's normal to have one side more effective than the other, this also changed the more I put her to the 'weaker' side). I was crestfallen to hear a friend could get 180ml from a single side in about 20-25 minutes. For me it was taking twice as long and getting nowhere near that much. 6-8 weeks in we are happily getting 150ml to 160ml in a single 20 session from one side, which I do first thing in the morning after her feed from the side I don't feed her on (as it's fit to burst after her longer stretch at night now).
Basically it will take a little while to get the expressing up, don't worry if it takes several weeks. Make sure you do it after a feed so you aren't empty when your DS gets hungry. It won't be instant - your body will ramp it up as you ask for it and as the feeding pattern becomes established - persevere!
I also find that if I don't express for even just one day, the next day there is a little less available, your body will react quickly to the supply/demand.
However, now at 13 weeks I'm only doing it once per week as she is through the night and it's far far easier and quicker for me to do all the feeds directly (plus I love feeding her and lucky enough to be able to EBF) and just remind her what a bottle is once in a while :)
Good luck!
p.s. anecdotally everyone I know who expressed (4 in my nct group) started doing it within a couple of weeks and had no problems doing so.
p.p.s keep using the lansinoh as your nipples may take a battering with the expressing to start with but you'll not need it forever, just a couple of weeks.