The reach to the bars should be comfortable. If it's too short you will be cramped on the bike. Too long and you will roll your shoulders forward to reach the bars, which will make them hurt after a while. Or you will ride with your elbows locked, which makes you unstable because you are twitching the bars every time you hit a bump.
Whatever you do, do not move the saddle forward (or back) to adjust the reach to the bars. That's the wrong way to go about it. If you move the seat forward then you'll be putting more weight on your hands, and you won't be able to pedal as efficiently. You set the seat position relative to the cranks, then adjust the handlebar height and reach.
Correct size frame is the start point. Even then there might be some adjustment to the bar stem length to fine tune. If you have too "Long" a top tube then a shorter stem can be fitted and likewise with a too short TT where a longer stem can be used. BUT-I don't like going more than 10mm either side of the original fitted as That would have been worked out by the manufacturer to give the best handling. Fit a far longer stem and more weight will be over the front wheel and conversely a far shorter stem will lessen weight. Both of which can affect the handling But "Can" possibly improve the handling for some.
Bar height is subjective. Just face it-We are getting older and need a more upright position on the bike to relieve back ache. One of the fallacy's about bike set up. Some of us have found that a low bar height to saddle works- some have found that a higher bar works. It is up to you to find out what you need and what is comfortable. What is important though is the saddle position to the pedals and height. Once you have that position it is fixed. Then there is the angle of the bars. I normally set them up so that I am comfortable on the Hoods of the brakes and that is just trial and error to see if you can improve or worsen the position.
In fact there is a lot of trial and error on bike set up. Once you have it nearly right then move the saddle 3mm at a time or adjust the bar height by moving spacers one at a time.
The other way out is to spend money on a bike fit- but this can be expensive and until you know what you want can be a waste of money and time.
How long was I in the army? Five foot seven.
I rather like Peter White's approach to getting started with a fit. If you read through his site, you'll gain enough information to make reasonable decisions and have the language to discuss fit with others. http://www.peterwhitecycles.com/fitting.htm
A conclusion is the place where you got tired of thinking. - S. Wright
Height of seat, length of top tube, stem height and length, angle, size and drop of handlebars, where your feet are on the pedals, crankarm length and size of frame all help to make a bike feel right. You have to adjust a little bit at a time to avoid discomfort. Peter White and the forum post mentioned above will both help with this. It's a long road to proper fit, and I find myself doing it every year as I age . For me, I start with the pedals and cranks, then set seat height, then worry about the bars and stem . Others will have different methods .
For me there are many goals. Mechanically, all the moving parts move freely, smoothly, and through their entire necessary range. Should feel like new. I should feel confident that all the bearings are properly clean and lubed.
Biomechanically, the bike fits me like a glove. When I get on, the saddle is precisely where my sit-bones want it to be, and all pressure from the saddle is in the right places (only!). The position will allow total freedom of leg movement for hours on end. When I reach forward to the bars, all positions should be easy to find with low hand pressure and no back or neck cramping.
Just a start ...