Tai Chi is an excellent exercise for your posture. While not free, Bruce Frantzis' book, Opening the Energy Gates of Your Body, gives a very nice explanation of standing and sitting alignments as it relates to Tai Chi, as well as the flow of Chi. (It costs less on amazon if you can find it.)
Since you are concerned about prolonged sitting with all its negative effects, you want to think about corrective exercises to stretch out tightness and strengthen weaknesses that develop over time, as well as prevention.
Prolonged sitting affects your posture negatively by tightening muscles that are in the shortened position when you sit (such as hamstrings, hip flexors, pecs etc.) and lengthens muscles that are in the rounded position (such as across your upper back like mid-traps and rhomboids). A muscle that is too short or too long is no longer at it’s optimal length, which causes it to weaken and lessens its postural support.
Exercises for Tightness vrs. Weakness:
To compensate for the prolonged sitting position at your laptop, think about the positions your joints and muscles are held in while you sit and design your exercise program to “undo” those positions.
Stretch flexors that tighten such as hip flexors, pecs, and hamstrings.
Cobra and lying backward over a ball help to stretch out your front side. (This question/answer gives some more information about stretching).
Strengthen back extensors, mid and lower traps, rhomboids, lats and core muscles. Something as simple as using resistance bands at your desk can help you strengthen some of these muscles. The seated rows using cables, or bent-over rows using weights are good for strengthening the upper back. Plank and bird-dog work your core and back muscles.
Using good posture while you sit helps to prevent multiple problems. Our site has some free info on sitting and standing posture. Additionally, you may want to check into the Alexander Technique, a posture and movement method, Pilates or yoga. Use a good ergonomic chair. Some people find that using an exercise ball intermittently as a desk chair or a balance disc in your chair helps improve sitting posture.
Spinal cord anatomy: Here is a spinal cord overview and diagram of the different vertebral levels, cervical, thoracic, lumbar and sacral vertebrae along with the nerve root levels. If you sit with a posture that compresses at certain points of the spine, you can irratate both joints and nerves.
Hope that gives you a good start. Also take a look at exrx's posture page for other corrective exercises. As you said it will take some time to get your own program set up, but to get started, try taking short breaks to do a cobra stretch (assuming you don't have a back problem), some bird-dogs, a few rhomboid contractions against gravity or with a resistance band, and a quick plank exercise.