The accepted 'best' method of training for a specific sport is to perform that sport - as far as muscle fibre recruitment is concerned, for cycling, the recruitment patterns are very specific, Muscle fibres have a 'memory' and as with any memory process, the more often the neural pathways are stimulated in the same way - the more efficient becomes the muscle recruitment.
To test this out, after a few months of decent cycling outings, try running hard - unless you've been a competitive/regular runner you'll end up with quite sore muscles afterwards (delayed onset muscle soreness) and feel like a carthorse while you're running !
As for cycling, the biomechanics of pedalling require the cyclical recruitment of Quads, Hamstrings, lower leg muscles, lower back muscles, abdominal muscles in a particular sequence to produce effective (i.e. power efficient) pedalling style.
Stair climbers per se are very useful for developing endurance in the group of muscles which allows you to perform that activity, likewise performing squats is excellent for producing explosive power over a very short time frame - not exactly replicated when riding a bike, although the benefits are useful in a racing situation.
If you are a 'relative' newcomer to cycling, then I suggest that the best approach is to ride the bike as often as you can, over different terrains (hilly/flat) for different distances and at different speeds:
Terrain - flat (ish) rides over long periods (3 H +) at a reasonable speed, but high cadence for aerobic conditioning and endurance.
Hills - varied between long steady seated climbs to improve muscle strength and endurance, and short sharp hills to develop explosive power.
Different speeds - related to terrain fast leg cadence to improve muscle fibre recruitment and muscle memory, slower cadence when pushing higher loads to develop leg strength.
At this time of year, ideally join a local club ride - most club rides are now adopting a 'wind down' to winter training rides which mostly comprise of long steady distance to build endurance for the arrival of next years racing calendar !
Good luck with the training, but don't get bogged down with specifics - ride the bike more and enjoy it - this time next year you should be a different rider - keep a training log !
cycling - since May this year. 
