Get stone cards to deny others and strengthen your position. Having stone cards release you from making moves to acquire from alterative sources (market, board etc.). If you have the most of the stone cards in a four player game, you can use it to build points while your opponents will be short for the larger projects.
Try to do all three Egizia “projects” even if you only do a little. This will get you 6 points per round which is 30 points. If you are going to miss one, make sure it is worth the 3 forfeited points. In the beginning you might just have to do this to secure important resources like stone or green fields.
The one rule for an Egizia beginner: always build on the Sphinx! This space gives you points for your discards and most likely extra points for completed cards that you keep. Even if you think you might not be able to fulfill a condition, keep it, you never know what will happen. If you can get combinations that focus on your current strategy even better (i.e. get cards that focus on the grave).
Keep an eye on your opponents and especial their grain and stone supply. You can adjust the flooding or scoop up a required card to keep them back.
If you are building on the pyramid try to get the bonus for the most on the row. It helps to build first and build the first stone of each row. A trick it so build the first few of the bottom row to ensure you tie, and then build the first few of the next row. Remember you do not have to complete the first row to start on the next one.
Grave points add up, but once you are passing 20, the extra will do you no good.