Delta is good for earning via their multi-partner reward promos and the occasional car rental, but as far as identifying it as "Skyteam", if it were me I would utilize Alaska for that purpose.
Remember: Friends don't let friends earn DL miles for flights. (infrequent flyers that is, FFs will have other advantages including elite status)
Credit your occasional Skyteam flight to AS or a dining for miles meal or Live Search redemption to keep your miles alive.
Redeem for Skyteam flights, or BA flights, or AA flights, or Qantas, or Cathay, or Air Pacific, etc. Hence, your miles you can utilize for skyteam flights are not limited to just skyteam and not limited to Delta's horrible booking engine and redemption rates.
For *A, US is a good option. You can get 25-35k for the Barclay's CC (they have a personal and biz and may be lightly churnable). You can earn via Live Search and Dining for miles as well. They seem to have a yearly multi-partner promo too. And they have a pretty good *A redemption chart.
As far as hotels go I would recommend SPG, as long as you have the SPG Amex CC. An infrequent flyer should be accumulating SPG points via regular CC spend (especially with the US Mint coin deal), then the points will be there for you when you do decide to travel and need a decent hotel room and/or need to xfer some SPG points to your 1:1 SPG/Airline mile program to get that flight you need.