Well, for the grocery thing, I suggest going at a time when the store is less full of people if at all possible. Also, have a list ready so you don't need to dawdle. If you can you can bring a safe person with you. That really helped me at first. Shop in small doses more often so you don't have to stay there so long. Have somebody waiting to pick you up so you can leave when you are done. Well that is how I got through it. Hope this helps.
As for things I haven't conquered yet.
Afraid of heights, really afraid.
Still nervous in crowds.
I don't take planes.
I don't drive.
I don't sit in the middle of a row at the theatre or the movie. I feel the need to sit at the end of a row. Oh and I don't sit at balcony levels or higher.
Doing my banking can send me into a panic.
I can't stand strangers in my house. Only people I am close to can come in.
... There must be more but I don't remember them hahaha!
This is the list of things I have conquered or accomplished.
I leave my house alone or accompanied.
I exercise.
I can shower with nobody with me in the bathroom, I can even close the door.
I can take nice long baths by myself in the room.
I can leave home to travel (I did this by bus).
I can take the bus, even long distances.
I can use public transportation.
I have worked.
I have collaborated on a paper that has been submitted for publication.
I have graduated from a bachelor's degree.
I go see movies and plays and shows.
I go to the restaurant even on Fridays and Saturdays.
I go to the mall on my own and shop.
I go for walks accompanied or not.
I can leave the house without taking my anxiety meds with me.
I can meet new people and have a nice time chatting with them.
I can go to the dentist and the doctors alone.
... I am sure there are more.
I added the list of things I have conquered or accomplished because I think it is important to acknowledge and celebrate all the big and little steps we take and all our successes even if we are not fully done with our "to do lists". Let's celebrate each success!