1: The WorldClim dataset has a lot of the data you want. It is free for non-commercial use and has interpolated 1 km resolution data on:
average monthly mean temperature (°C * 10)
average monthly minimum temperature (°C * 10)
average monthly maximum temperature (°C * 10)
average monthly precipitation (mm)
2: For rainfall you can download the TRMM datasets: http://trmm.gsfc.nasa.gov/
3: The PRISM Climate Group's data is exceptional. Their raster products include precipitation, max temp, min temp, dewpoint and historic data.
NASA's MODIS site has a wealth of data as does this USGS site. You will find a wide range of products there from vegetation indices to emissivity and burn data.
I should add some context to the use of satellite to retrieve rainfall. In descending order of accuracy, they fall into satellite based radar (some of the TRMM instruments), microwave sensing, and Infrared/visible (MODIS).
Radar - This senses the hydrometeors directly and is very similar to ground based radar (with all the built in assumptions that you should read about), but will suffer from narrow swath width, hence infrequent repeat time. Because of this it's only good for statistical studies, such as climate.
Microwave - over oceans it senses hydrometeors by emission, and is nearly as accurate as radar. Over land the background is too uneven for this to work well, so is based on scattering of ice particles, so that warm rain is not captured. It's less accurate than the ocean retrievals. Also it currently has a resolution on the order of 10 km. Multiple satellites mean you can get retrievals every couple of hours on average. There are algorithms such as C-morph that uses IR tracking to interpolate between microwave satellite overpasses.
IR/visible - highest resolution, but the accuracy is poor. The general feeling is that it's better for statistical work (climatologies) rather than instantaneous rainfall, but if you have nothing else it at least gives you a sense of what's going on. Read the comparison to gauge data very carefully - you'll find the error bars will be large for a single event.
For all of these, don't be fooled into thinking a long term average provides the accuracy for a single event. This is a common mistake newcomers to remote sensing of precipitation often make. Part of this is the point nature of gauge measurements and the almost fractal nature of rainfall; part is the inherent inaccuracy and statistical nature of the remote sensing technique. Try to find a scatter plot of individual events versus retrieval to get a feeling for what's really going on.
For long-term climatic studies, precipitation data from Global Precipitation Climatology Project (GPCP) is also a good choice, which has daily and monthly data sets at a global basis.
Data is available at: http://precip.gsfc.nasa.gov.