Yes drowning ... it was referred to by early NZ explorers as the "NZ way to die". Your options are numerous! I assume you haven't been to NZ before but great choice, Asp Park is often drier than the west coast side of the alps or Fiordland regions with Wanaka itself in a rain shadow (which care lure you into thinking its OK in the hills). Wanaka has a huge DOC (info) centre with all sorts of information maps, weather, SAR details etc which should be your first port of call. Walks in NZ are largely dependent on weather.
Some ideas ... (If you can, get a copy of Classic Tramps in NZ! Great pics! and inspirational trip ideas and will look great on the coffee table) Always difficult to gauge someone's abilities, but in NZ, and for that matter probably anywhere, start within capabilites then progress as one builds knowledge on aspects unique to NZ - like weather, rivers, wind, reaction to local beers, SANDFLIES! culture etc etc.
Some suggestions which I'll list against a rough level rating which is personal, only you know your capabilites. The higher the level, more challenging, less people but mother nature doesn't differentiate and the scenery is spekky on any.
Level 1 Great Walks - well signposted, tracks maintained (by NZ stds), bridges, support at huts (warden,gas,even flush loos!). Popular, you have to book and you pay! Your Routeburn is one that time of year.
Level 2 - still have hut wardens, markers and trails (less maintained) can be a bit more indistinguishable eg Rees-Dart, Cascade Saddle, Wilkin-Young, Greenstone-Caples.
Level 1 & 2 are generally supported by transport to and from.
Level 3 - bit more remote ie more camping, navigational skills as tracks are less distinguishable, few bridges, no wardens. eg Five Passes, Steele Creek/Death Valley
Level 4 - basically step before alpine. remote, animal trails, little human sign. Little support, axes/crampons
For short trips out of Wanaka, either hitch or bus out of town to Raspberry Creek carpark then to French Ridge or Liverpool. If longer, go to Aspiring Hut then over Cascade Saddle and out via Rees or Dart (I prefer Rees for Mt Earnslaw) but transport can be limited). Transport out and stay in Glenorchy (great old pub and BEER). Transport from town will take you to start the Routebourne or Greenstone Caples (limited supplies in town available - go to Qtn to restock). O/N option up the Earnslaw Burn. Many will do the routebourne then complete the round trip back to Glenorchy via the Caples or Greenstone (good trout fishing with hunting licence) or end it and bus back via Te Anau from The Divide (more tramping options here). If you want more of a challenge, on a not so travelled track (Level 3), head over Steele pass from the greenstone to the upper caples hut (so not doing the McKellar Saddle) then hop over the range east again (think this is part of death valley, option Emily Pass back into routebourne) dropping back to Glenorchy area (might have to hitch out). Tent needed. Other one you mention is Wilkin Young out of Makarora. Level 2. Limioted supplies from Makarora. Be sure to include the side trip to Lake Crucible on the Young side. If you're chasing a Level 3 option extension, turn R at the Forks Hut up to the rabbit pass and out via the East MAtukituki to hitch back into Wanaka. Some might think this teetering on a 4. Try to include an O/N at Brewster Hut further W from Makarora. Though the hut has changed, still has superb views.
Good Luck!