To Lina, Valgaav doesn't pose too much of a threat near the beginning of TRY. Later, she realizes how powerful he is and becomes a little more wary of him. Valgaav mainly wants Lina dead to avenge Gaav.. little does he know, that it wasn't Lina who killed Gaav.
Gourry Gabriev
Gourry fought Valgaav only a few times that I can think of. In the beginning of TRY, Valgaav underestimates Gourry's ability [probably because, well, he's Gourry, he looks stupid naturally!], and Gourry becomes very offensive fighting-wise, probably because Valgaav thought so little of him.
Zelgadis Greywords
Zelgadis and Valgaav don't interact much. Most likely, Zelgadis thinks of him as just a pothole in the road to finding his cure. They never actually fight, but of course, they are enemies.
Amelia Wil Tesla Seyruun
Like Zelgadis, Amelia and Valgaav don't interact very much at all, but Amelia does seem pretty wary of his power. She noted that Valgaav must be very strong because he is the subordinate of the foe that Lina couldn't defeat in the past.
Xelloss
Putting it lightly, Xelloss and Valgaav HATE each other. One of Xelloss' tasks in TRY was to try to sway Valgaav to the Mazoku side. Of course, Valgaav hating both Ryuuzoku and Mazoku, disagreed, so Xelloss then tried to kill Val. That didn't happen, though...
Hellmaster Phibrizzo
Phibrizzo and Valgaav never actually met, but if they did, it's most likely that they'd HATE each other. Phibrizzo and Gaav were very distant from each other and there was a lot of friction between them, so it'd probably be the same for Val, too. And, of course, since Phibby killed Gaav, that's even more of a reason for Val to hate Phibby.
Demon Dragon King Gaav
Gaav was the one who changed Val into a chimeric Mazoku/Ryuuzoku. For keeping him alive, Valgaav was very grateful to Gaav and would give his life up to him, in fact, in TRY, avenging Gaav was Val's sole purpose for living.
Filia Ul Copt
At first, Filia hated Valgaav. But then she started to see the truth behind Ancient Dragons and Golden Dragons, and felt pity for him. Of course, it wasn't exactly likewise for Valgaav, because hated Golden Dragons for the genocide committed on his kind.
Gravos
Gravos was saved by Valgaav somehow, and therefore became servant to him. Gravos doesn't have all that much respect for Valgaav, probably because Valgaav has NO respect for Gravos and treats him like dirt!
Jillas
Jillas respects Valgaav wholeheartedly. Valgaav saved Jillas' life when Red Foxes were being killed, therefore Jillas is the last Red Fox. Valgaav doesn't interact with Jillas much, and Jillas takes orders from Gravos, mostly.
Almace
Almace hired Valgaav to find the Weapons of Light for him in return for Lina's death. Almace is from the plane that Darkstar comes from, and a Shinzoku too, therefore Almace and Valgaav don't totally get along. Of course, Valgaav respects him in some way.
Darkstar
Darkstar doesn't actually show its personality [if it even has one] but Valgaav wanted to harness its power to destroy the plane in which Lina lives on, to have the whole world start anew. But of course, in the end, it was Darkstar who killed Valgaav..