Yes,「ている」can be used to talk about a repeated action, but this is one of the tricky grammar points where Japanese and English diverge and the nuances are hard to grasp. Since you already use 「毎日」in that sentence, you can just use the dictionary form. When you use 「ている」to talk about a repeated action you stress its continuity and the fact that it is still ongoing and you are still doing it regularly/habitually.
In your case, it is not necessary to use 「ている」. Also you are talking about going to the store every day, so logically you come back after shopping. A good way to say this is
You can change them to 敬体/です・ます if you want. It only changes the politeness level.
And when you find yourself in a situation where you do need to use 「ている」, you can just attach it to the last verb in the sequence and use the te form for preceding verbs.