Why did Riot Remove Stealth Wards?
The big idea behind Riot removing Stealth Wards was to make vision more of a carefully thought-out tactical decision and less of a ward spam-fest.
Our goal with vision is two-fold: first, we want to keep pushing the vision game as a team effort and, second, we want to make vision more dynamic than days past. - 2016 Preseason Blog
General Strategy Considerations
In terms of strategy, you should primarily be aware that vision is a finite resource. All stealth wards are now on some kind of cooldown. Many players still place wards with reckless abandon, not considering the fact that they'll need them for a team fight in the jungle in 45 seconds. Or that they'll need their trinket up to help secure Baron or Dragon.
Wards to Watch
The major player in the current patch is the blue trinket. It defies the ward cap, allows you to place wards at an extended range, doesn't expire, and on top of all of that, has a lower cooldown than any other trinket. The obvious drawbacks are that it can't hold charges, it isn't invisible, and it only has a single hit point. Regardless of the drawbacks, this trinket is incredibly valuable for scouting brush and getting deep intel without putting yourself at risk.
The next biggest player in the current patch is the Vision Ward. This guy is awesome right now. Sure, you can only have one, but they now cost only 75g. Fantastic cost-effectiveness. This is the only way to see invisible champions at the moment. The red trinket only gives you Rek'Sai-esque foot-step pings for invisible champions. Buy pink wards. You don't really have a reason not to anymore. Unless you don't have room in your inventory.
Finally, the Rift Herald is a bit lackluster at the moment. It's expensive in health investment to actually kill it, and its benefit is fairly marginal. Riot did mention in a recent blog that Rift Herald will be getting a tune up Soon™.
Rift Herald's currently looking a bit underutilized and we believe it's still too expensive (in terms of health lost) for many champions to take. We don't want to make the Herald a mandatory objective, but do want to see it used more. - 2016 State of the Preseason Blog
The possible ramifications of a valuable Rift Herald are simply more considerations for prioritizing ward locations. Riot is forcing us to carefully choose what we want to be watching for, and when.
Going into Season 6, you should be more carefully considering where you want vision. Whether you need it now, or whether you'll need it later even more. Don't let this choice paralyze you, but try to think a step ahead to the next Dragon or Baron fight. Make sure the wards you drop have a specific purpose. Vision is precious; make it count.