Södermalm is probably the best area in Stockholm. I recommend taking the metro to Slussen and taking the Götgatan exit. You can also walk down from Gamla Stan which also brings you to Slussen. Then stroll southwards while making sure not to miss the side streets on both sides.
Mosebacken is connected to Södra Teatern, which is a theatre and club/bar. Mosebacketerrassen is open during summer and has a great view.
This place has a 100% vegan buffet. It is situated a bit higher and has a great view from the garden where you can eat. Be sure to bring a valid student card if you have it because the student discount is in the 40% range. Food is amazing and in summer there is also a grill outside.
It's a brewery and pizza place. You can find their beers in the Systembolaget and in other bars as well. It can get a little crowded so they do not always have space. They don't do reservations either, but there are plenty of alternatives nearby so you have alternatives if they are full.
Another beer place and bar. It is located directly at Götgatan. They have a big selection of their own beers.
The best coffee ever. Drop are a roastery and have this cafe. The cafe is right outside Mariatorget metro stop. Their coffee tastes great and is very traceable (on the Drop website you can usually find what they paid the coffee farmers). The pastries they serve are from Svedjan bakery. Go there, pick a bean you like and enjoy. There are a few seats outside, and a computers and are only allowed in a separate room. Also recommended to buy their beans (online).
Kajsas is an open air bar in the park.
A nice bookstore with novels, photobooks, magazines, etc.
Chutney is a full-vegetarian place. Prices are great for the dagens: 110 Sek for lunch, 119 for dinner. You can usually get a second plate and even mix two dishes. Highly recommended for some good affordable food.
A rocky hill that is nice to hangout at and enjoy the view.
Grandpa is a Swedish clothing/other stuff store. It has slightly pricey but high quality brands, it's hard to describe, just go there. There are a few stores in the city but the one on Söder is definitely a nice one, and close to other highlights.
Boulebar is located in the far south of Södermalm in an area called Tantolunden. You can swim at Tantolunden
An amazing Mexican lunch and dinner place. The food is just so good. They have multiple restaurants all over the city but one of the better ones is the one on Söder. It has a few nice seats outside.