The inns and cathedrals are both basically the same mechanic: double or nothing. putting an inn-road on a road makes it a double or nothing road. It also means that if the road is not completed at the end of the game, the road is worth 0 points. Putting a cathedral in a city does the same thing to the city. Double points if it completes, 0 if it does not. The same basic strategies apply to both the inns and the cathedrals.
If you draw an inn or cathedral and think you can include it in a city or road that you can complete, control, and defend, go for it. You’ll score a lot!
If a road or city that your opponents are playing in appears to be worth a lot of points, but is difficult/impossible to complete, put an inn or cathedral in there. Now not only are their meeples stuck, they also lose whatever partial points they were counting on.
If you see an inn or cathedral that is going to score, and you have no part in it, it is often worthwhile to spend tiles trying to make it impossible or difficult to complete. It is also maybe worthwhile to spend meeples to join in, or completely steal it. All those other players who did the early work spent their turns giving points to you! Stealing an inn or cathedral takes just as many actions as stealing a big city or road, but gets you twice as many points if you succeed.
Cathedrals are worth insane numbers of points, but are very obvious. There are also not many cathedral tiles. They are also very difficult to complete because they are city on all four sides. You probably need some help to score a cathedral, and you need to start working on it in the early game. If you can partner with another player (who is worse than yourself) to complete and share a big cathedral, you can effectively eliminate all other players who were not scoring in that cathedral.
A cathedral tile drawn in the late game is usually going to be spent making someone else’s incomplete city worth 0 points. Unless the perfect spot somehow magically exists for it to go in.
Inns are also worth a lot more points than people realize. You could say they are inn-conspicuous. A normal 3 point road would normally be overlooked. A six tile long road is considered a long road, and would demand some attention. A road with an inn is half as long, but gets about the same attention from other players as a road worth half as many points. Be sure to pay close attention to inn-roads. They are worth more than you realize.
As in normal Carcassonne you want to have an active road and city at almost all times. That way if you draw a tile you can reliably convert it to points. However, because of chances of drawing an inn, it is a good idea to have an active long road going. If you have a 5 tile road and suddenly draw an inn, you’ve now got a 12 point road (the inn tile is still a road). Just make sure it’s one that can be safely completed and can’t be stolen easily.
The expansion also adds a big meeple. This is a single meeple that is as strong as two meeples for the purposes of breaking ties for control of shared cities/roads/farms. You’ve only got one big meeple, so you want to use it wisely. Use it on an inn or a cathedral that is going to complete successfully to make it difficult for opponents to steal it from you without also committing their own big meeple.
Get the big meeple into things late, not early. If the big meeple is trapped on the board, opponents have time to see it and overcome its strength. You want the big meeple to go on and off the board frequently, so you can extract its full power. Try not to use it for things that are not contested, otherwise it’s just acting as a normal meeple.
This means don’t put the big meeple in a farm early. You want to use it on cities and roads. But then at the end of the game, you absolutely have to get it into a high scoring farm to make sure you extract its value.
The expansion also adds the ability to play with a sixth player, and adds many more tiles to the set that are not inns or cathedrals. Even when there are less than six players you don’t reduce the tile set. Make sure you reconfigure your calculations with regards to how easy it will be for things to finish. Players are going to have more individual turns, more tiles per player, and the game will be longer. There are also a few rare tricky tiles added to the set that are very valuable. Complex cities and roads are more likely to eventually finish and score, especially if they were started early in the game or have multiple players cooperating. Do not forget this.