Not ever test nation shares a specific rivalry with one particular nation. So based on who a test nation (the fans, players, media hype) would to beat the most, or be most excited about.
India-Pakistan and the second greatest cricketing rivalry are at the top. Nothing more to add about those two.
After that
England - South Africa (In recent years, they have beaten South Africa away, lost at home and every loss led to a resignation of the test captain).
Australia - South Africa (similar to England, lost 3 consecutive home series to South Africa, but managed to beat them away.) The last few years have seen some of the most riveting test cricket and pretty much each series has had at least one classic batting or bowling performance.
India - Australia. The 2001 series created this and the last few years saw India do everything to become Australia - arrogant, bullish and disparaging towards the rest. Every series since 2007 have seen some controversy at some level. With Kohli now no longer friends with Australia, you watch this space for the next series between these two brats.
Pakistan - England. Often overshadowed by bigger rivalries each of them are involved in but this has always had the makings of a great rivalry - history - Pakistan won their first ever overseas test and drew the series 1-1. 87 saw Pakistan's first ever test series win in England and Imran's peak as captain and allrounder. 92 Final, 92 test series - Wasim and Waqar, reverse swing/ball tampering, 2000 Karachi series. Pakistani players are always eager to play in England and they love playing in England too given the fan support they get.
New Zealand - Australia - sibling rivalry - younger brother beating the older brother
Sri Lanka - Based on online chatter, Sri Lankan fans love to beat India
West Indies - Lot of history with England in the past with the 'grovel' comment and the racial/colonial overtones of the 60s and 70s. Australia is another one with the Frank Worrell Trophy and the 99 series demonstrated just how much it meant to both sides. Not sure if West Indies have a meaningful test rivalry with anyone now.
Bangladesh - India - this deserves a post of it's own. I will come back later for that