Beowulf's quest for glory

The purpose of Beowulf's quest for glory is initially to establish a name for himself.

Beowulf is a nobleman, with many relatives who are kings and a father who served in Hrothgar's court. Beowulf's father is Ecgtheow and his grandfather is also named Beowulf (changed to Beow in some versions). In lines 262 to 266 he refers to his father as a "noble vanguard-warrior".

The point of him being a nobleman is to make him a more appealing hero. The noblemen were supposed to be heroic and great with many deeds to their names and be the great leader that could effortlessly get rid of any enemy.

Beowulf is not the leader of the Geats at the beginning of the poem, but he is a member of the court of king Hygelac. So he is of noble birth and is essentially a knight at the time, with a claim for being King of the Geats at a later point.

Obviously, Beowulf is aware of his lineage, because he spends a fair bit of time discussing his ancestors and what they did. He spends time discussing what he did as well, but before he fought Grendel, he mostly just talks about defeating ogres and opponents in battle, which, while impressive,  is rather vague. The ogres and opponents are never named. "I defeated 5 ogres" is not quite the same thing as "I defeated the fiend from hell, Grendel, who terrorized the meadhall Heorot". Named opponents are more impressive.

Beowulf is fighting Grendel because Grendel is widely known as a terror, and he wants to be known as the one who defeats that terror.

There is an interesting bit where Beowulf is speaking:

I have also heard       that the evil creature

 in his recklessness     heeds not weapons; 

then I it scorn --      so that for me Hygelac may be

my liege-lord        blithe in his heart-- 

that I bear a sword     or broad shield,

yellow-rim to war        but I with my grip 

shall fight with this fiend

Beowulf states that he will fight Grendel without weapons to make Hygelac "blithe in his heart". Blithe means happy or joyous. Part of the reason is for personal glory, but these lines (433 to 439) also make it appear that he is fighting to bring glory to the Geats and to Hygelac. This makes sense given that in Scandinavian society, ideas of individualism weren't as common as they are today.

Later on, when he is King, he fights the dragon in order to defend his people. (lines 2312 to 2399 is it being discussed how Beowulf became king and him making his decision to fight the dragon.)

Overall, it appears he fights for glory, and while the glory is partially for himself, he seems to be fighting for the glory (and, regarding the dragon, safety) of the Geats as a whole.