Steve Rogers/Captain America is in perfect physical condition as a result of the super serum he was injected with. This took him from a small, weedy person unfit for military service into who he is today.
His mental strength is a combination of the serum and his own beliefs and convictions. Leading up to the super serum experiment, he is shown as standing up for what he believes is right, regardless of whether standing up for them means he gets beaten up. Those ideals are what got him chosen above all others when it came to the experiment. The serum amplifies those aspects, just as they enhanced Red Skull into the strong but ultimately bad person he was.
From those two paragraphs, we can deduce that he has had his physical strength since the serum injections, and his mental strength from before then. His mental strength is part of who he always was. However, he has always had a large amount of physical endurance, which is partially (at least) down to his mental toughness. In the first avenger when he's being beaten up for standing up to someone talking in the cinema, he's repeatedly punched to the ground but gets up saying:
I can do this all day
He can tough it out because his mental strength allows him to keep going. The serum enhances that. This mental strength gives him the belief he can do a lot with the body he's been given, and therefore allows him to push on through different injuries, like when The Winter Soldier shoots him on the helicarrier.
Arguably he has had side effects from the experiment. His enhancement is a side effect of it, and his increased mental toughness is also one. Whilst that was the intended outcome, he also has to live with knowing he's likely to outlive his friends due to an enhanced healing factor and the fact he was a "Capsicle" for years after The First Avenger took place.
Due to the relatively short time which has passed between the films, and what they depict, we don't know if there are any other side effects of the serum. The initial batch did have issues (causing Red Skull's, well, red skull) but that had been worked out by the time Steve Rogers has it administered. Being put on ice for 70 years or so also means that any decay in his powers could have been halted along with his ageing.
There's a lot of things which could happen and be seen as side effects as The Avengers timeline continues, it could be that the effects haven't had time to manifest themselves yet.