Spend, spend, spend!
Governments have spent quite a lot for space exploration using public money.
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Spend money (on someone/something)
The preposition you need in English is spend on:
Governments have spent quite a lot on space exploration using public money.
(examples here)
Spend money (doing something)
Another way to express this is using __ing:
Governments have spent quite a lot of public money exploring space.
Spend money (somewhere)
You can also spend money in a particular place:
Governments have spent quite a lot of public money in developing countries.
In this case, of course other prepositions of place will be possible - around, at, etc.
Spend time
The language used to talk about spending money also applies to spending time, except that we don't often spend time on someone, rather we spend time with them:
Last weekend I spent some time with my girlfriend. We spent Saturday evening at the cinema and after that we spent some time walking along the beach. She noticed that I had gained weight and urged me to spend more time on physical exercise.