Can anyone advise me of the best way to do Darjeeling-Sikkim-KTM regarding transportation between all 3 places, which order would be best, etc.

The best way to go to darjeeling-sikkim-ktm is to book any travel agency program,
There are lots of travel program to visit that three places.
Last time my friends and I went to these destinations,we booked a travel plan.
Tour Duration: 13 days
Destination(s): Darjeeling-Sikkim-Kathmandu
Specialty Categories: Art/Architecture, History Tours, Museum Tours, Cultural Tours, Mountain views, Hiking & Trekking
Season: Round the Year.
Tour Starts at: Darjeeling
Ends at: Darjeeling.

Some small group adventures to take in the places you want to see.

I believe it's currently (and has been for quite some time) impossible to go from Sikkim (or, indeed, directly from Darjeeling) into Nepal.

From Sikkim you'd need to backtrack through Darjeeling to Siliguri, then get a bus to the Raniganj border crossing from where you can cross to Karkabitta in Nepal. From Karkabitta it's a looooong (on the order of 12 hours) bus trip to KTM.

Alternatively, you can get a bus to Hetauda and try to get a share jeep over the mountain road to KTM. This will shave 3 hours off the journey, and it's a beautiful trip, but there's no guarantee there'll be a share jeep (or any jeep at all) there to take you when you arrive.

When are you planning to go?

Last year I flew from Kathmandu to Nepal's eastern border on Yeti Airlines (fairly cheap), crossed over the border via one of the minibus taxis that are at the little airport, and picked up a share jeep (just me and 11 Indians) for the 2 1/2 hour ride up the mountains to Darjeeling. After a week there, I took another share jeep further north to Gangtok, Sikkim. Spent another week in that region, including a 3-day jeep tour to the far north. There's a heavy military presence up that way due to the border with Tibet/China, and as far as I recall, it's not possible to cross the border in that region.