Congratulations on starting a regular exercise regimen and working at maintaining a healthy weight. This is one of the very best decisions you can make for your health. Cardiovascular activity like walking and going up stairs (more challenging than walking on a flat surface and a great way to add
more difficulty to your workout) are a great start.
The symptoms you describe of tightness in your Achilles tendon are likely coming from the added work that your calf muscle are doing while going up stairs. When your muscles contract--as they do when you are using them--they get shorter. In the case of your calf muscles, this places an extra 'pull' on your Achilles tendon which is what anchors your calf muscle to the bones at the back of your ankle.
In order to avoid muscles strains or sprains or pain after your workouts, stretching is an important part of any fitness regimen. If you have not been stretching regularly, this would definitely be a good thing to add to your routine. A personal trainer may be able to give you some good pointers in person. In addition, when it comes to the calf muscles, the shoes you were are work may also contribute. Wearing heels causes the calf muscles to shorten which may further exacerbate the 'stretch' you feel after climbing stairs. Adding some daily stretching should help take care of this and let you achieve your weight loss