The first movement of a symphony shares the same form as the first movement of a sonata, a piece for a solo piano or a solo instrument accompanied by piano. Allegro means fast, so sonata-allegro means a certain form, played quickly.
The sonata-allegro form begins with a tune. In Beethoven's Fifth Symphony, it is the dah dah dah daaaaaaaah sequence. Then it goes to a contrasting tune that has a different sound because the home pitch is different from the first tune. Finally, there is a closing tune that brings the whole thing to an ending. It's like the end of a sentence in the middle of a paragraph. The two tunes and the closing theme comprise the exposition, which lays out the materials that will be used in the rest of the movement.
In the development section, the composer takes snippets from the tunes in the exposition. One snippet might be played several times using different pitches.
Finally there is the recapitulation, which is like the exposition, with one difference. When the second theme comes along, it remains in the original key instead of going to a different home pitch. The closing theme comes along and the first movement is done.
In the second movement, you get to hear the composer's best melody-writing and you can enjoy the feelings of a beautiful song.
The third movement is in an old dance form called the minuet. The minuet is in waltz timing (ONE two three ONE two three). There is a feeling of happiness about a minuet it has one tune for 8 measures, which is repeated, then another tune for 8 measure, repeated. It moves to the trio, where you have two more 8 measure tunes that are repeated. Then, you move back into the minuet tunes..
The last movement can have any number of different forms it could be sonata-allegro again or it could be rondo form. Rondo is ABACADAEA, where A is a main tune and all the other alphabet letters are contrasting tunes. It's a fun form to listen to because it's interesting to hear how the composer will create a new contrasting tune after each repetition of the A tune.
When you first listen to symphonies, it may be hard to identify all these features. I would recommend listening to Haydn and Mozart because the elements of form are clear in their works. After you have listened for form many times, hearing the form won't require so much concentration. Then you will be able to hear a symphony, maintain your interest, and enjoy it.