Best way to west side of Mt. Adams, WA from I5 southbound

167 and 512 are freeways, effectively a beltway around Tacoma. But you need to pay attention to lanes. If you don't you end up staying on 167 which Ts onto 161. Which isn't a disaster, since the 512 exit to 161 isn't that much further south. But the 31st exit from 512 is meant to be the main access to south bound 161.

For either 7 or 161 I'd try to exit from 512 rather than I5.