What's going on in "I have been to the store many times"?

I thought been was the past participle of to be, but it seems to behave like the past participle of to go in this case:

I go to the store every Wednesday.

I have been/gone to the store many times.

*I am to the store every Wednesday.

Also

I go through the tunnel.

I've been/gone through the tunnel.

*I am through the tunnel.

What's happening?

'been' is the past perfect tense of 'be'.

Been is often used as the past participle of come and go.

1. Have you ever been to London? (= Have you ever visited London?) [Visit = to go and stay for a short time.]
2. I have been to see my uncle. (= I have paid a visit to my uncle.)

Note that been is used for completed visits.

Notice the difference here:

1. She has been to the store. (= She went to the store and returned.)
2. She has gone to the store. (= She is at the store at the moment, or perhaps, on the way but didn't return yet.)
3. I have been to the store many times. (= I visited the store many times.)
4. I have gone to the store many times. [this is a present perfect tense, 'had' in place of 'have' makes it past perfect tense.]

[subject + auxiliary verb(has/have) + auxiliary verb(been) + main verb(action+ing)] is present perfect continuous tense.

Been is also used for present continuous tense. There is usually a connection with the present. An action that is stopping or has stopped recently:

1. I'm tired [present] because I've been running.
2. Why is the grass wet [present]? Has it been raining?
3. You don't understand [present] because you haven't been listening.

[subject + auxiliary verb(had) + auxiliary verb(been) + main verb(action+ing)] is a past perfect continuous tense.

If you are using the present perfect to make a general statement about the places you have visited (as in the OP's examples), then both gone and been are possible.

If, on the other hand, you are referring to one current instance, then there is a difference in usage and meaning:

• She's been to the store

means she went there and is now back.

• She's gone to the store

means she is not here now.

By this token, the statement:

• I've gone to the store

cannot be said to someone standing next to you. It would only be possible as a written or phone message.

The verb to be to meaning to visit a place and return from it can only be used in the perfect tenses.

In general, when you use to be, it means that you have gone and returned. When you use to go, it means that you have gone and are still there.