What details in Beowulf reveal the importance of pagan warrior values, such as a belief in fate, a taste for boasting, a pride in loyalty and a desire for fame?
The details mentioned in the other answer to this question are insightful, and they point to important examples of warrior values in the text. To add onto these ideas, I'd like to focus on the value of loyalty. Pride in loyalty is a major value and theme in Beowulf, and it's exemplified in several ways. However, one of the easiest ways to determine the importance of loyalty in the text is the importance of gift-giving. Just as a warrior was required to provide his king or lord with fighting services, so too was the king or lord supposed to honor this loyalty by giving sumptuous gifts. Thus, when Hrothgar lavishes Beowulf with gifts on two occasions (first when he defeats Grendel, then when he kills Grendel's mother), Hrothgar is recognizing and honoring the warrior's loyalty and service in battle. Thus, a king shows that he is proud of his warriors' loyalty by giving many gifts, while warriors show their pride in their loyalty to their lords by bearing said gifts (these gifts might have included rings, weapons, armor, etc.). By extension, a king with many loyal followers would have been obliged to give many gifts, and so a generous king would have been seen as blessed with his people's loyalty. Based on these details, we can see that the pride of loyalty is an important element in the text, as it cements the relationship between lords and their warriors.
Since the Anglo-Saxons did not believe in an afterlife, the only way they saw to achieve immortality was, as you said, to achieve fame. This would ensure that their stories of heroism would be sung in the mead halls for future generations--as indeed Beowulf's is, or you wouldn't find his tale in your textbook. This desire is what leads Beowulf to leave his home to challenge Grendel, and it is what leads his fourteen men to accompany him.
In Chapter XXI, Beowulf says, "Each of us must await the end of his path in this world, and he who can, should achieve renown before death! That is the best memorial when life is past and a warrior's days are recounted," reflecting his belief in fate as well as his beliefs about immortality.
Beowulf's vocal joust with Unferth is evidence of his love of boasting. He proves in this section that he is as equally skilled as an orator as he is as a warrior. He exaggerates his feats almost to the point of humor; for example, he claims he swam for many hours under water in chain mail armor. This boasting only adds to Beowulf's mystique and further promotes his legendary status among the Geats and the Danes.