It is currently not possible to move from a 1-60 (Classic Era) server to a 1-70 (TBCC) server. All TBCC realms are scheduled to be progressed into WOTLKC (1-80). The available options are to either roll a toon on Classic Era or TBC (soon to be WOTLK). Best of Luck finding a place to play and congrats on the kiddo!
As far as Vanilla goes, theres been a number of attempts to consolidate their much smaller playerbase. The Classic Era realms have been ‘connected’ (Blizzard’s term for a merge) based on timezone & server type (PvE/PvP).
You will find the majority of Classic Era players concentrated in the NA West for PvP and NA East for PvE. This is because Blizzard has established free transfers for Classic Era toons onto destination realms in these Clusters: Whitemane for PvP and Mankrik for PvE. (Free Character Transfers Now Available -- WoW Classic Era)
Some friends and I remain on our native realm cluster in the NA PvP East (Benediction/Faerlina/Heartseeker/Incendius/Netherwind). Our cluster is smaller but we actively do dungeons and are helping the lowbies gear up. Battlegrounds for Classic Era are Cross-Realmed, meaning that every Classic Era player is in the same queue for instanced PvP. These typically only fire off on Fridays/Weekends. If you have any questions, ask away!