Can at be the first word of a sentence?

Yes, "at" can be the first word of a sentence for a grammatically correct sentence. Usually, at is used to start a subordinate clause. A subordinate clause is the part of a sentence that isn't correct all by itself. An example of a subordinate clause is "At my house". Notice that the previous sentence is a fragment. It is a subordinate clause. However when you add an independent clause, (the part of a sentence that is correct all by itself) you have a grammatically correct sentence. An example of a grammatically correct sentence that includes a subordinate clause is "At my house, my friend lost his tooth". In almost all cases, at can be used to start a sentence. Use your common sense or ask a teacher, etc. to find out when using "at" to start a sentence is incorrect. For your help, I will write some examples of when using "at" to start a sentence is correct, and when it is incorrect.

Correct:

-At a mountain I saw an experienced hiker climbing without safety equipment.

-At soccer practice I scored a goal.

Incorrect:

-At the park. (Fragment)

-At the football game.

(While the two incorrect sentences could be used as answers to questions, they are fragments, which makes them grammatically incorrect.)

Having trouble with my description of subordinate and independent clauses? I have given you some examples of both to clarify. The words in italics are the subordinate/independent clauses.

Subordinate Clauses in Sentences:

At a mountain I saw an experienced hiker climbing without safety equipment.

The subordinate clause in the previous sentence was "At a mountain".

Independent Clauses in Sentences:

At a mountain I saw an experienced hiker climbing without safety equipment.

The independent clause in the previous sentence was "I saw an experienced hiker climbing without safety equipment".

Note that in most cases, English teachers would prefer the independent clause to be at the beginning of the sentence as this makes the sentence clearer to the reader.