FYI there is also a Rugby code called Rugby League (13 a side) which has slight differences to the Rugby Union. Its probably more comparable to Gridiron than Union is. As an example in League you have 6 plays to try and score in Union its unlimited possession.
Club Rugby
The strongest club competition is in Europe. And that's mainly because Covid basically killed off the Super Rugby which was made up of the Kiwi Aussie and Saffer teams. The Saffers left after Covid and they joined the European competiton. And none of the other South Hemisphere countries club teams outside of Australia and New Zealand are miles behind in level. Its like a Div 2 or 3 College playing an NFL team. Super Rugby was electric the skills the pace the physicality the athleticism was different compared to European Rugby.
New Zealand domestic club rugby is probably the highest level in terms of skills athleticism and violence (hitting). Then after that I'd say South Africa and France have the 2 next strongest domestic club competitions. Usually the French teams win the European Club competitions as they have the most money to spend on talent. While Australia is a powerhouse in Rugby Union but have been in slowly decline for the last decade both on and off the field. It is also the 3rd sport behind Rugby League and Aussie Rules Football so the popularity in Australia has taken a hit.
In Rugby Union International Rugby is King. So the best Rugby competition outside of the World Cup is the Rugby Championship which is between New Zealand Australia South Africa and Argentina so basically the 4 best/strongest South Hemisphere teams. This year competition starts on 10th August and runs till end of September. After that ots the 6 Nations so England Scotland Wales Ireland France and Italy. Which runs from 1st weekend in February till the end of March. And then you have the Autumn Internationals which is the Southern Hemisphere teams touring the Northern Hemisphere 6 Nation Teams which take place in November and early December.