Some tips from the top of my head. I tried to make a list but I think it would be very misleading so I decided to just cover some helpful examples:
First, nothing is worth getting hit with Hard-Hitting News. If you have to choose between a potential HHN and trashing a key asset, let the Corp have it.
Second, no card trash is worth ruining your economy. Don’t spend a lot of money trashing cards that could go into making runs. Your goal is not to play the asset game but to prevent the Corp from playing it while you run.
Commercial Bankers Group: Any political asset must be destroyed as soon as possible. No exceptions. There’s a list two of them are banned and the other is on the limited list. If you let people have Bankers, you lose.
Corporate Town: This will crush your economy and make you unable to play. There’s a big exception, though: Never, ever trash Corporate Town if it would put you in a hole.If it’s in a big remote, consider ignoring it and running HQ instead. You won’t recover fast enough if you run that remote and the Corp will score.
Daily Bussiness Show: You won’t beat an asset spam deck by playing their asset game, you’ll win by keeping them in check long enough that you can steal agendas and win. DBS prevents agendas from getting to HQ and gives them more assets so it should be a priority target. However, don’t
Hostile Infrastructure: This is the most important and powerful card for Genomics and other Jinteki asset builds. You’ll lose if you have to trash cards with them on the table or if you stop trashing cards because they are on the table. There’s nothing worse than ignoring them just to hit an Snare! in HQ and realize it will be there next turn. They are expensive, though, so you can wait to destroy them. But go and destroy them.
Team Sponsorship: Trash this card ONLY after trashing other important targets. If you keep the Corp poor and can run on centrals, this card won’t be great. Trash them when you can “clean the table” or leave the Corp with a few, unimportant cards.
Drip economy cards (PAD, MHC, Illegal Arms Factory, etc.): A PAD Campaign takes 5 turns to recover its investment, 4 if you spent a click to check it. This means you can leave them on the table for a while. However note that if you don’t trash these cards, you’ll eventually lose hard. Simply keep an eye on the Corp’s economy and trash these cards before the Corp’s advantage becomes too strong.
Turtlebacks: This is a higher priority target than a PAD because it can give 2 credits per turn. Often decks that use this card rely on it quite strongly.
Mumba Temple: Like Turtlebacks, it’s a higher priotity target because it gives 2 credits per turn. Despite its power, it doesn’t give “real” money so I prefer to trash the Turtlebacks first.
Melange, Mining Corp: No longer used very much, but trash it on sight if it is. It’s a one-card economy.
Jeeves: For me this is a low-priority target if the Corp isn’t scoring. Getting one extra install or an extra credit and not even on all turns is not a big worry. I mostly treat it like a PAD.
Amani Senai: This is a card that punishes you for being poor. Focus on having money first, then focus on other important targets. Remember, it only fires if someone is scoring and you can control that. Trash it when you are comfortable making runs.
Hope that helps