Which of the gold training is most efficient, in terms of strength per gold?
Basic training is +5 strength/0 gold. Q1 training is +2.5 strength/0.19 gold. Q2 training is +5 strength/1.49 gold. Q3 training is +7.5 strength/1.79 gold.
Q1 training is the most efficient paid training, at 0.076g per 1 str. Q2 training costs 0.28g per 1 str. Q3 training costs 0.238(6) per 1 str. Q2 is just horrible, Q3 is good if you've got money to burn.
You can also upgrade your basic training facility to boost your free daily training. Q2 will give you 10 str/day, and Q3 will give you 15 str/day. 1st upgrade costs 20g, 2nd - 50g. It's a good investment - 70g in upgrades will buy you the equivalent of 5+2.5+7.5 training, which costs 1.98g/day, so it'll pay for itself in 35 days.
The most efficient one is the free training ground - Weight Room raised to Q4. Then Climbing Center Q4. Finally Shooting Range and Special Forces Center (both Q4) - they are basically the same when it comes to cost efficiency.
You can also use this website to calculate costs etc.