The NSWRL only 3 had venues 1908-09, and any given club could've been allocated any ground - Showground, Birchgrove & Wentworth.
Fair to say though Balmain had Birchgrove, Glebe had Wentworth, Souths & Easts had Showground, and everyone else had no home ground. Newcastle had 3 games in 1909 at Newcastle Showground.
In 1910 Norths gained NSO, Newtown got Metters, Wests got St Lukes (Concord).
Wests lost St Lukes 1911-12 & were again homeless til Pratten Park in 1913-14, then St Lukes 1915-19, then back to Pratten again.
Newtown had Metters 1911-12, but were then all over the shop, finally settling at Marrickville from 1925.
Norths were again without a ground 1916-17 (presumably due to WW1 needs) & 1923 (not sure why, but have recollection of reading that someone forgot to renew the Council lease and the RU beat them to it). Norths held some games at Chatswood Oval mid 1930s.
Souths varied across the 3 Moore Park venues 1908-48, then Redfern 1949 onwards.
Easts varied across the 3 Moore Park venues, though had Trumper Park in 1915.
Balmain had Birchgrove, used Drummoyne Oval during 1932-34, and the latter season was also first games at Leichhardt.
Glebe had Wentworth Park, but had no home ground for 1920-24.
Annandale generally had Wentworth Park (shared with Glebe), but in final season (1920) had Birchgrove (shared with Balmain).
St George gained Earl Park in 1925, before that they were not settled, though some in that first era were at Hurstville Oval.
University could've played anywhere, but Pratten Park does appear quite frequently for their home games. As stated earlier, Uni did not get to train or play on the Univ oval or ground - the only exception to that was in 1930 when the Australian Universities RL team played the touring NZ Kiwis (who won 18-12).