We surely need to differentiate three regions for Ecuador. There are the famous Galapagos islands. There is the highly elevated Andes area. And there is the Amazonian area.
As you can see from my recommendations on the left side, defining the best time to travel is quite hard.
Since the country is eactly located at the equator, there are not the typical four seasons.
Mainly there is rainy and dry season but sometimes not even that.
cold season (rather dry): June to November
warm season (rather wet): Dec/ January to May. The warmer season is better for diving and there tend to be more young animals around.
dry season: June to November
rainy season: December to May.
dry season: August to November.
wet season: December to July. Especially in June and July it is very humid and it is hot.
Ecuador is a developing and rather poor country.
That implies that the health system is not up to European, American (etc) standards.
You should check with a travel doc. Tell him or her to which areas you want to go. The sierra is not as sensitive as the Amazonian area.
If you go to the Amazonian area yellow fever and Hepatitis A (+B) should be a topic.
There are places in Quito and other bigger cities where you should not got. Listen to the locals.
I didn't and I got in trouble. People around helped but still better not to get in the situation.
Moreover watch your belongings. Again, I wasn't careful enough and they stole stuff out of my backpack on a bus ride. Contact me if you want to read my stupid story. ;-)
Galapagos is different. It is remote and more touristy, safer, I would say.
You can get everywhere by bus. Just as the locals.
But it takes time. Especially going from the sierra to the Amazonian area.
Galapagos: rather high
Galapagos: check in advance
obviously there is Galapagos. Great for diving.
On mainland Ecuador there is the nature. The Andes are incredible.
So is the Amazonian rain forest.
And there is history and culture like Quito and Otavalo.
If you want to go to Galapagos and you do have the budget, DO it!
Don't trade it for anything. I did and it was a mistake. Today I am convinced there is nothing like Galapagos. I went to another island and it was disappointing.
Go and see Quito as well as (the native market of) Otavalo as a day trip.
Wanna hike a volcano? Go to Cotopaxi.
I went to Puyo by bus and worked as a volunteer in an Animal rescue center "near" Tena called Sacha Yacu.
You should absolutely go to the rainforst and do a hike there. The forest is great..
It might also be nice to see Cuenca. Check out the free walking tour (freewalkingtourcuenca.com) or their facebook page.