Loads of great rides, here's some ideas. Much of this is well known to experienced riders but I thought I may as well be comprehensive! There might be a couple of lesser known rides in there.
Bike paths: There are two down south, the Veloway which follows the southern expressway from the start at Darlington down to Noarlunga, and the Coast To Vines trail which starts at Marino (you can follow the railway to get there), intersects the Veloway at Reynella Interchange and again at Noarlunga and then goes on to Maclaren Vale and Willunga, so there's various combinations of the two trails that be done. The Coast to Vines follows an old rail alignment so is generally flat, the Veloway has a bit more climbing though nothing super steep. If you go all the way to Willunga then there is the Old Willunga Road hill climb, a good challenge. It has been replaced as the main road so fairly quiet for traffic.
The Crafers bike trail - this follows the South Eastern freeway and then up the old highway to Eagle on the hill and Crafers. Quite a long climb though mostly nothing really steep, but takes a good steady effort. From Crafers you can cut through some back streets and then head on up to Mount Lofty.
Quiet Roads up to the hills: Norton Summit is the classic entry level ride to the hills, there are two roads up there, Norton Summit Road is the windier one which is an easy ride and doesn't have too much traffic since Old Norton Summit Rd is more direct. I always take that one to avoid the traffic. At the opposite end from entry level is the turnoff to Woodlands Way which is one of the most ridiculously challenging climbs in Adelaide (I never made it all the way to the top, on my best effort I came to a standstill about 10m short).
Montacute Rd is a more challenging ride which ends up at Norton Summit. As it isn't a direct route to anywhere further in the hills it only has local traffic, but also as a key route into a fire-prone area it is well maintained and a good road surface (in particular the bottom part gets resurfaced after flooding). It's deceptively long, just when you think the climbing should be over you turn a corner and somehow it is still going up. A good rule of thumb is that the climbing isn't over until you see the hobbit house!
Piggot Range Road - this is a quiet and relatively gentle climb into the hills from down south, good way to Clarendon.
Less quiet roads up the hills: Gorge Road is a little busier but still not too bad. Fairly gentle except for the climb up to the dam, then it's really nice through Cuddlee Creek
Belair Rd is a good climb, fairly busy but there's a bike lane and it's not too bad (though I once had a narrow escape on the way down, closely followed by cars, when two magpies were fighting on the road right in front of me!). You can stop for the view at Windy Point and then continue up to Belair, and the National Park.
Greenhill Rd is spectacular either going up or down, but is busy, I tend to only do it when it's likely to be quiet eg. early Sunday morning.
Bains Road - a slightly more direct way up to Clarendon from down south, but a bit harder and busier than Piggot Range Rd.
Challenging Climbs Apart from Willunga Hill already mentioned ...
Corkscrew Road - connects two of the routes mentioned above, Gorge Rd and Montacute Rd. While the twists and turns famously give it its name, the straight part leading up to it is what I find the toughest.
Pound Rd - you can either go down Pound Rd from Montacute, or from Norton Summit head down the road into Basket range and then along one of a couple of routes that get to the intersection at the bottom. Lovely quiet part of the hills and a good tough climb out. If you do the descent it can be dicey, I've seen people run into trouble there.
Flagstaff Hill - busy but the road has been duplicated and there are better bike lanes, though I've not been up since then. Good climb to get up to Aberfoyle/Happy Valley area which can then lead to other destinations.
Clarendon - Clarendon is a great place to visit, but when it's time to leave any way out is going to involve some climbing. Some are tougher than others.
Belair National Park - there are lots of gentle rides around the bottom of the park (and accessible by train for those who want a nice park visit without climbing up), but at the back there are some good climbs, there's Cherry Plantation Road which comes out at Upper Sturt Rd, or Saddle Hill Rd which comes out on Sheoak (which then leads to Crafers). From Upper Sturt Rd there is a great ride down the Sturt Valley and back up (with a fair bit of climbing) to Crafers, or you can head out through Ironbank and Cherry Gardens to Chandlers Hill (and back down through Aberfoyle Park) . To ride up to the park you can use Belair Rd, or Shepherd's Hill Rd (busy but with a good bike lane), the ridiculously steep bike path from Lynton Railway station (it gets to about 20%), or come in by one of those back ways. There is also Old Belair Rd, I tend not to do that one because it is busy and not so bike friendly.
Pole Rd - the corkscrew of the South. From Ironbank Rd it goes down into the Sturt Valley then winds up to Upper Sturt Rd, close to an entrance to Belair National Park.
Other Nice Roads Some other favourites connecting various points once up in the hills.
There's a good way between Mt Lofty and Ashton - down the summit rd to Greenhill, then along Ridge Rd, then either Woods Hill Rd down to Norton Summit, or keep going straight and connect with Marble Hill Rd and so on to Montacute.
Basket Range is a nice area when you keep going past Norton Summit. It's a fair climb back up though! Also connects by Fox Creek Rd to Gorge Rd, or by various routes to Greenhill around Uraidla.
Aldgate Valley Rd down to Mylor is lovely, you have to find your way around backstreets of Stirling to get there. Once at the bottom it isn't too far to visit Hahndorf - though watch for tourist cars! Haven't been there for years, but I recall there is bike path out along the main road through Oakbank, Woodside etc. From Lobethal you can get back to town through Norton (with a fair bit of climbing) or the Torrens gorge vie Cudlee Creek (easier)
Thought of some more:
North - don't get north too often these days, forgot about Paracombe, which is a really nice area. Can get them via a turnoff on Gorge Rd which is this steep uphill turn on a bend which looks daunting but isn't as bad as it looks, or up lower north east rd. Watch out for kangaroos.
South - for a good challenging climb, from Hallet Cove Station, follow the road down to Hallet Cove Beach, then climb back up the other side. You end up in the industrial area at Lonsdale, you can get to the Veloway from there.
Also, if in the Noarlunga area, a detour down to Old Noarlunga is quite nice.