Can you recommend some basic stretches and/or yoga poses to correct posture? My neck leans forward and my shoulders round forward (very badly) from 15 years of working in an office, staring at a computer screen. It has really caused my upper body to round forward badly.
It is very possible to significantly alter your posture by practicing yoga and you may find that improving your posture also leads to other beneficial side effects such as better sleep, more confidence, and more energy. This is a highly worthwhile undertaking, good for you for starting it.
It is going to take more than a few simple stretches to correct bad posture. In order to create significant changes in your body, you are going to have to make a commitment to this change. You will need to learn new ways to align your body and then do this alignment periodically all day as you remember. I know this seems like a lot of work, but it is so worth it. If you know there is something you can do to become a better version of yourself, you owe it to yourself to make this commitment.
I recommend taking an alignment-based yoga class (Iyengar or Anusara) at least twice a week.
OR even better, do a private yoga class with an alignment based teacher once a week, once a month, whatever you can afford. This could be in addition to classes or in addition to a video.
A video should not substitute for an in-person teacher as you need the individual feedback to make sure you are doing your exercises correctly. For a video, I recommend "Yoga to the Rescue" with Desiree Rumbaugh. Even if a class is not possible for you right now for whatever reason, order this video, it will be extremely helpful.
As far as a few stretches to get you started, do each of these for 1-3 minutes while breathing deeply and slowly through your nose.
Stand with your back to the wall. Press your shoulders and the back of your head into the wall. Be sure that you are not pressing your neck into the wall, there should be a slight curve away from the wall with your neck. This should feel really good. If you have the room, you can also slide your arms up and down along the wall--reaching your arms up as you inhale and down as you exhale. Another variation, in this same position reach your arms overhead -- interlace your fingers and turn your palms face up.
Turn to face the wall and reach your arms up the wall. With your arms straight press just your fingertips into the wall (the palms of your hands domed away from the wall). As you press your fingertips down, your arms will engage and the outsides of your shoulders will lift away from the wall. Keep this action and soften your chest and forehead towards the wall.
Walk your hands down the wall and your feet away from the wall until your body is at 90 degrees. Hands press flat into the wall, shoulders and hips right inline with your hands, and feet right under your hips. As you press your hands into the wall, lift your outer shoulders up towards the ceiling and at the same time soften your heart towards the ground. Make sure to keep your head in line with your spine and not let it drop since your neck already slopes forward, you don't want to encourage that.
Lay flat on your back, bend your knees and place your feet flat on the ground, approximately under your knees. Hug your arms in by your sides as tightly as you can and then bend your elbows so that your fingertips point towards the ceiling. Press into your upper arms and the back of your head into the ground as you arch your upper back away from the ground. Your low back will come up too, but not your hips.
All four of these exercises will feel really good. Because they use the resistance of the wall and the ground for feedback and stability, they will be more effective for you than poses where you are not stabilized in this way.