Reading books together as a fun, relaxing activity is one of the biggest proven factors in early reading ability. Do you read for fun? Is this something that your child would see and want to emulate, just like wearing Daddy's hat or playing with cell phones and tools?
The teacher's games and alphabet cookie ideas are spot on. Make the activity its own reward: reading is not something you do for other people, its something other people help you learn because it opens up so many cool things you can do on your own. Imagine sending your name on a rocket to the moon, or writing letters on the beach as a message to God.
If either of you gets frustrated, take a short break. Come back to it soon with a full belly, empty bladder, and fresh patience. Treat homework as a natural part of life. It keeps happening, so find the fun: just like taking a bath with toys and warm water, or making dinner with a smile. Try to quit on a good note, and remember that's just right for next time.
A 6-year-old has plenty of time to learn to read. Maintaining the desire / curiosity, and cheerful discipline, is the best help you can be. Think "Mary Poppins" not "Mommy Dearest."
We knew a homeschooling kid whose parents never made them do anything until they were interested. Their oldest was a little willful, and didn't try to learn reading until she was past 10 years old - I think she was 11 or 12 when they moved across country, and she couldn't email her friends like her little sister was doing. Her new friends were not super impressed that she couldn't read, either, and peer pressure plus a can-do attitude from home life got her started in a big way. By age 13 she had taught herself to read, and started a monthly newsletter for her friends in distant states. She keeps that newsletter going, it's been 3 years now.
Now at age 16, she's attending college courses.
All it takes is wanting to learn, and knowing that everything gets better with practice. So keep things fun, read great books aloud, play games, and check in from time to time in case there are other things going on at school that might be hurting the intrinsic desire and curiosity. Kids can be mean, and not all teachers are as experienced or creative as the ones who have responded here.