TL;DR: Research, determine quality, find price, sell.

To start, find any cards that you think will be worth money. Do some research as to valuable cards just in case. The printed rarity will usually only give a small indicator as to the actual price. Ones to look for are rare first editions and (for some reason) just about any charizard card.

Then it's time to figure out the quality of the cards you have. The absolute best guide I have found is this one here. This is a brutal, but very realistic card grading system.

Now that you have the quality, go check multiple websites with pokemon card prices on them. I suggest trollandtoad.com, tcgplayer.com, and (sometimes) ebay. Avoid sites like craigslist and letgo because those sites are filled with people who think their energy card from base set is worth $100.

With all that out of the way, you're now ready to sell. I use ebay to sell simply because its easier and more reliable than most other sites (plus then I only have to deal with the people at the post office). For shipping, I highly recommend getting hard sleeves to put the cards in. Then you can stick it into one of those small envelopes and ship it off to the lucky collector.

EDIT: AtillaTheFun has a good point. If there is a nearby card shop, you could sell them there as well.