You can keep your pepper seedlings in the peat pellets, to help minimize transplant shock (they are especially good for that, since you can plant them pellet and roots together in the ground). If you do not plan to transplant them out directly in the garden for several weeks, you might consider at some midway point between now and then, transplanting the seedling (including its peat pellet) into a larger pot. Otherwise, wait and plant outside directly, after hardening off. As to the lighting, seedlings usually need about 12 to 16 hours of light. Make sure that your desk light is not on constantly overnight—use a timer. It would be better to use a grow light that offers the spectrum needed by plants, or a fluorescent shop light. The windowsill and desk lamp (if using a regular light bulb) may lead to thin, elongated stems. If there is no other option, though, adjust the lamp so that it is close to the seedlings, but not too close to cause any heat damage. Keep rotating your plants every day, even twice a day, so that the stems do not grow in one direction (toward the source of light). Good luck!
You can keep your pepper seedlings in the peat pellets, to help minimize transplant shock (they are especially good for that, since you can plant them pellet and roots together in the ground). If you do not plan to transplant them out directly in the garden for several weeks, you might consider at some midway point between now and then, transplanting the seedling (including its peat pellet) into a larger pot. Otherwise, wait and plant outside directly, after hardening off. As to the lighting, seedlings usually need about 12 to 16 hours of light. Make sure that your desk light is not on constantly overnight—use a timer. It would be better to use a grow light that offers the spectrum needed by plants, or a fluorescent shop light. The windowsill and desk lamp (if using a regular light bulb) may lead to thin, elongated stems. If there is no other option, though, adjust the lamp so that it is close to the seedlings, but not too close to cause any heat damage. Keep rotating your plants every day, even twice a day, so that the stems do not grow in one direction (toward the source of light). Good luck!