You can jump start the rock curing process by rinsing in freshwater and giving it a bleach soak, but the true cure doesn't begin until it goes into saltwater and starts to leach the death from within. This is typically monitored by tracking nitrates and or phosphates, as these two guys are an easy test compound coming from decomposition. Here's a good video from BRS, but I'd highly recommend skipping any acid dip. My opinion is that the biggest thing acid washing does is give you less rock, and is extremely dangerous:
If the rock is fairly clean, the cure typically takes about a month. If "dirty", expect two months or more. The issue, if you don't cure, is that the rock continues to leach nitrates and phosphates, increasing the likelihood of algae blooms in the tank.
When it now comes to cycling, you need to minimally test for ammonia (but if you wish to see the whole process, include nitrite and nitrate). What's critical about the cycle is that you've developed enough bacteria to quickly consume (convert) ammonia. Ammonia at very low levels is damaging/deadly to fish. So what you do is start the cycle by either putting in something like a table shrimp or actually dosing ammonia. Once you get to zero ammonia, I'd suggest testing by dosing actual ammonia and then test a little later to make sure it's not there.
Hope this helps