The Internet Scrabble Club (ISC) is the best place to play Scrabble online. You can compete against anyone from beginners to world-class players, and there is a nice rating system so it is easy to find competitive games no matter how good you are.
My username on ISC is jdscheff. If you want to play me, it's probably best to send me a message somewhere and pick a time, because I usually only go on to play a random game every few weeks or so... my skills are slowly atrophying, I know. This is evidenced by my rating of 1132 (as of January 2017), which is well below my peak of 1261.