I can't say that I've played until my fingers bled, but once I developed a small blister! The experts say that playing shouldn't be painful, "no pain no gain" doesn't apply here. If something's really painful or your fingers are getting numb, stop before you cause yourself permanent damage.
One great piece of advice I was given years ago was to get a guitar stand and leave the guitar out so you'll be more likely to pick it up and play it. Keeping it in a case can keep it out of sight and out of mind. However, there are many situations, like if you have young children or live in a small place, where leaving your guitar out might not be practical.
As for strums, I often use a basic folk/rock strum that works well for a lot of songs. I recommend starting by practicing 1 2&3&4&, strumming down on the downbeats and up on the upbeats: down, down up, down up, down up, and repeat for several measures. After you master that, try playing the same pattern but NOT strumming the strings on your down stroke of 3 (sometimes called a "silent strum" because your hand keeps the same down up pattern), strumming 1 2&-&4& using down, down up, - up, down up. When you're learning to change from one chord to another, you might want to leave the last "&" strum off the pattern and play 1 2&-&4 (down, down up, - up, down) to give your fingers a half-beat head start to form the chord so they'll be ready on 1.
Another strum that works well on a lot of trad songs is one I like to call the bluegrass strum (note that it's not a true bluegrass rhythm guitar strum, but you can buy a DVD if you want to play bluegrass authentically). Start by practicing 1 2 3&4& (down, down, down up, down up) over and over, then silent strum on 3 and play 1 2 -&4& (down, down, - up, down up). After you master that, learn to pluck just the bass note on 1, so that your pattern becomes bass, down, - up, down up. I use this pattern on songs like Lukey, The Night Pat Murphy Died, Jack Hinks, Old Black Rum, and Captain Kidd, among others.
Hope this helps, and have fun playing!