Temples in Tamil Nadu and elsewhere are important in their own right. That said, there are various temples that are also grouped together for any number of reasons. There is always some legend or history associated with these groupings.
To me personally, the order of visit to these temples is not important. However, there are people to whom it is, and this page lists out some of the important (and also some lesser-known) temple groups. Each temple group has its own page with more information about the grouping, and also the list of temples forming part of that group.
Paadal Petra Sthalams are the 275 Siva temples revered in the Tevaram, on which one of the 3 Nayanmars called the Moovar – being Appar, Sambandar and Sundarar – have sung at least one full pathigam.
Vaippu sthalams are temples mentioned in the Tevaram verses of the Moovar, but only by reference, and not a full pathigam. Different sources provide varying number of Vaippu sthalam temples.
Divya Desams are 108 places (including 106 earthly temples, plus Tiruparkadal and Vaikuntam / Paramapadam) revered by one or more Azhvars, in the Divya Prabhandam.
Arupadai Veedu are the six temples / abodes of Murugan, which are considered battle stations. Of these six, five are on hills / hillocks, while Tiruchendur is by the sea.
Navagraham temple groups are sets of nine (sometimes 10, including Sani’s son Mandi) temples dedicated to Siva or Vishnu, which are associated with the nine planetary deities.
Sapta Sthanam temples are sets of 7 temples regarded as a group. Each group often has common or connected festivals / customs. The Tiruvaiyaru Sapta sthanam is the most famous of these.
Sapta Vitanga sthalams are 7 temples associated with Muchukunda Chakravarti and the story of the Maragadha lingam.
Pancha kaa kshetrams are 5 temples in Tamil Nadu, which are located in what were forests (kaa is a corruption of kaavu, meaning forest).
Pancha krosha sthalams are temples within a distance of five kroshams (about 10.4 km). There are five main sets of such temples.