Like Audren, I thought the strategy had to involve doorbells as that's what I had the most luck with as well as insanely quick upgrades at seemingly random times. It never occurred to me that different toilets would have differing upgrade times, so I never bothered with those once I built the small room with the one and only door on my sim's property. I do not have Seasons or Showtime.
Otherwise my strategy was pretty similar to many others - I scored both a Wish Enhancing Serum and a Midas Touch elixir towards the end of the first week, so once I got to level 5 in handiness and upgraded the fire extinguisher that weekend, I quit the firefighter career, used both elixirs and got my LTW along with the motive mobile. Beyond the firefighting career objects and the times I was at the station, I stayed with doorbells until I maxed handiness which is when I started doing the unique upgrades. Once the unique upgrades were done I went back to doorbells.
The first LTR was the motive mobile, followed by the food replicator and teleporter (purchased at the same time), prepared traveler and, eventually, steel bladder. Traits were handy, inappropriate, kleptomaniac, ambitious and overemotional - in hindsight, I should have used loner or loves the outdoors in place of OE, and kleptomaniac ended up being irrelevant after the first week. I also spent a little too much time fulfilling wishes that had nothing to do with handiness or money. She was a witch so that she could conjure her own food, and early on I had her learn just enough alchemy to make her own invigorating elixirs. I took 4 trips total - one three-day trip to China, and 3 more trips for a total of 15 days in France (3 days then 2 more trips of 6 days each). I didn't bother with Egypt because there were no unique upgrades there and base camp doesn't have any doors.