Times are tight in this economic climate, and it is often easy to visit museums to learn about a country. As two separate sentences they are fine: Time are tight in this economic climate. It is often easy to visit museums to learn about a country. However they don't seem related and so you wouldn't join them together as you have above. If they are related in subject it could work, for example: Times are tight in this economic climate and entertainment can be expensive. However it is easy to visit free museums to learn about a country. I hope that has helped a little and not confused you more. Mandy

The sentence is grammatical. In terms of meaning, the two clauses do not have related meanings and so the reader would wonder why you have put them together in the same sentence. The phrase "the times are tight" is a cliché and so you have to be careful about how you use it. It relates to the economy and so to use "economic climate" is almost saying the same thing. In my experience, it is used with a specific group of people in a short sentence like this:. "Times are tight for investors in pension funds: rates of return have fallen...."