I want to be able to play rhythm guitar without learning how to play each chord used in a song when I am learning the song.
The best I am hoping for is a free website which provides a list of chords.
Just typing "Guitar Chords" into Google brings up some really useful sites with examples of chord shapes.
But it is also worth thinking about which chord shapes are worth learning first, too. Aside from learning the easiest chords (C, G, Am etc.), which include open strings and are usually on the first couple of frets, it is worth learning some movable chords, so you can use the same shape for every key. These are often, but not always, barre chords. (Importantly, though, movable chords don't have any open strings, which is why they can be moved up and down the neck to use in different keys.) Also, I would suggest that some chords are more useful to learn first than others. This seems like a good starting list:
This is a subjective list, and would only really just get you started, but this seems to be a pretty standard set of chord types for getting into Rock and Pop playing. The world of chord types, chord voicings and guitar chord shapes is vast, so you would quickly want to add to this list.
Just one other thing; I reckon it is worth learning two movable shapes for most chords to start with. You can play each chord type on all twelve pitches with just one shape, but knowing two will stop you jumping around the neck so much.
Even when you have learnt a wide variety of chord shapes, you will still sometimes need to learn chord shapes that are specific to songs; this is fine, sometimes very individual chord voicings give a guitar part a distinctive sound.
Eventually you will want to be able to work out your own shapes too; by understanding what notes need to be in any particular chord type, you can work out shapes with a sound and feel you like. But, you are absolutely right that learning a few shapes for the most common chord types is a really important part of learning to play the guitar.
To expand a little on Bob's superb (as usual) answer. He says two shapes of barre chords. Specifically, E and A shape work well on guitar. The E, taken up one fret, and the index finger substituted for the nut, gives Fmaj. By taking off the finger on the third string, you get min. By taking off the finger on the fourth, you get dominant seventh. Logically, taking off both of those, you get min7th.
The other shape, based on A, taken up one fret, and barred as above, gives Bb. And yes, you can play all the strings - some prefer leaving out the 6th, but when changing from one chord to the other on the same fret, it's easier to leave the barre finger exactly where it is.
On that A shape, move the second string down one fret, you get Bbm. Taking the third string finger off gives Bb7, so doing both will arrive at Bbm7. The maj7th is found by taking the third string down one fret. So they're all pretty east to find and move from one to another.
Finally, all these shapes can be moved up the neck, so you can play in a myriad of keys.