You get 5 base points per eligible spend just for providing your World of Hyatt membership at check-in (free for anyone to join). Again, HRC maintenance fees and taxes don't count but incidentals during a stay like food purchases would. This is true regardless of your tier with the Hyatt program (from base Member to Globalist). Starting with the 2nd tier (Discoverist), you get bonus points on base points earned. 10% for Discoverist, 20% for Explorist, 30% for Globalist. So basically a Discoverist would get 5.5 points per dollar, Explorist 6 points, and Globalist 6.5 points. If you get the Hyatt credit card, you are automatically a Discoverist. This and the free Category 1-4 hotel night annually is enough reason to get the Hyatt card and pay the $95 Annual Fee. Once you are a Discoverist, you can link your Hyatt Account with AA, and start earning Hyatt points with American flights (in addition to the usual AA points earned).
The next choice is how to pay the folio at checkout. If you pay with the Hyatt credit card, you get 4 Hyatt points per dollar (so this is why the card is sometimes advertised as 9 points per dollar for Hyatt spend --- it's 5 base for the Hotel stay + 4 extra for paying with the Hyatt credit card. Or you might pay with another travel card, such as the Chase Sapphire Reserve, which would net the 5 base Hyatt points + 3 Ultimate Reward points. Some might consider those 3 Ultimate Reward points more valuable and flexible than netting an additional 4 Hyatt hotel points.
If you pay with the Hyatt credit card, those 4 points will post automatically as it goes through the credit card. For the other points you may need to call World of Hyatt customer service to get them to post (after waiting at least 10 days after checkout). My understanding is that the HRC / Hyatt hotel systems are different and this is why HRC points may not post without contacting Hyatt, even if people have provided their membership number at checkin.