Planning to purchase a road bike I am 57 years old, 182 cm and weigh 105 kg. I have been cycling for the past 5 years, mostly with a city bike, Trek 7.5 FX. After a trial of a long distance lasting a week with that bike, I realized what it meant to be more aerodynamic. What would be the best option for me? More of an endurance type or more of a performance type of bike, such as Trek Madone 5.9? I tended to avoid bending forward while cycling too much until now. However, seems that I need to. 56 cm versus 58 cm? Any experience or thought would be very much appreciated. Many thanks in advance.
You basically get what you pay for, so there isn't any secret formula. Trek is one of the largest manufacturers on the planet, and is probably as good as any other on the market.

But have a look at, and test if possible, other bikes as well as Trek, because different models have different geometries, and another geometry might feel better to you.

Size also is only really judged by testing, but 56 or 58cm seems a good place to start from what we know about your height. In truth there is very little between these two sizes, so it is unlikely it would be a straight "fit"/"not fit" scenario. More likely one size will feel a bit more comfortable to you than the other. If two frames do feel absolutely the same, the smaller frame will tend to be a small amount stiffer and lighter.

Often with bikes you'll face a trade-off between a better frame or better components. Of course, it depends how averse you are to upgrading the bike after the initial purchase, but be aware that many components can be upgraded later (although it isn't very cost-effective) but upgrading a frame is usually prohibitive - more likely than not you're then in the realms of buying a new bike.