What should we purchase before we even arrive in Paris?
If you are talking about things like museum and transit passes to use in Paris, buy none of them until you get there. If you are talking about train tickets out of Paris, you'll get the best prices the sooner you buy them, on many routes.
What about things like Eiffel tower?
You should buy your Eiffel tickets as soon as you can. They sell out very quickly.
Keith as others have said purchasing ET tickets in advance is important, but not something to stress about if you didn't..they only sell a few in advance for specific time slots the rest of us wait, or go up stairs, or book tours with skip the line privildges.
If you are not going to buy a Museum Pass , and plan on visiting Versailles, that is another thing you could buy in advance if you chose to . Most everything else buy when you get there.
Eiffel Tower tickets now go on sale 3 months in advance and sellout quickly. If you are traveling with a group of two or more, I would book tickets on the first day your desired date goes on sale. You book by time of day and should arrive 15 minutes before the time scheduled.