I need weather and climate data for several countries.

Weather data is mostly taken locally, not nationally--for example, it rains differently from one part of a country to another, and it is measured at particular points.  Think, for example, Seattle versus Death Valley.  So it may not be helpful to assign a single number for each climate statistic to a country.  That said, you should be able to find data for meteorological stations in a country.  And there are, as I understand it, ways to deal with climate across an area, maybe using grid-based data?

Try these sources:

Global Precipitation and Climatology Center: https://www.dwd.de/EN/ourservices/gpcc/gpcc.html

European Climate Assessment and Dataset: https://www.ecad.eu/

NOAA’s National Climate Data Center also has some free global products—scroll down to get past the US-only stuff.

For a more general guide to weather and climate data, try the World Meteorological Organization.

The weather services of individual countries may also be able to provide data.