Adding front derailleur vs upgrading the rear one on a (ladies) bicycle What I want to achieve is going from having some gears to having more gears. To be more specific I would like to get more gear ratio range in both low and high end.

First of all I tried to look for the answer over several days(few hours at a time). I also hope this is not too open ended - I did read on how to ask a question and what not to ask. I'll try to be as specific as I can.

I did find several topics on adding gears to a single speed bicycle. Like these:


Whats the easiest way to add gears to my single speed
How hard is it to add gears to a bike that previously had none


Also found people going from multiple-gears to a single gear:


How to convert a multi-speed bike to single-speed with coaster brake? 


And MTB specific talk on giveing up the front derailleur:


Why single chainring is enough for trail and enduro riding


In several threads people have suggested getting a second bike but this is not a good solution for me at the moment.

The bicycle (Ammaco Classique):


Shimano RS-35 6-speed index revoshift gear lever
Shimano TY18 rear 6-speed derailleur
Gears:


Front single 42T cog
Rear a 6 speed freewheel with 14T-28T (G.Falcon 8BO). It measures 33.5 mm tall when rested on it's back on a table top and measured to the tallest point. 



I can usually handle most of the repairs myself and consider this a fun exercise. So I probably won't have to count shop hours in the overall cost. That said there are still several jobs/tools that I have jet to learn.

What I already know about adding front derailleur


I might also have to upgrade the rear one to match the gears
I'll probably need clamp-on cable guides
I'll most likely need to get a new chain guard
Also might need a new bottom bracket


What I already know about upgrading the rear one


I will probably need to get a new chain


Hope this is enough to get the discussion going. Sorry for the broken links - will fix them when I get sufficient rep.
Your only economically viable option is to get a new 6-speed rear cluster that has a wider range. Adding gears onto the front that bike is going to be almost expensive enough that it's probably not really worth doing. And adding more gears to the rear is effectively impossible. That's an overstatement, but it's waaay more money and work than you want to get into. 

To find a compatible rear cluster, you just need to figure out if your bike has a freewheel or cassette. The venerable Sheldon Brown has a great article about what the difference between the two is and how to identify which you have. Once you know which you have, just count the teeth on the largest and smallest rings. Then you can buy one with more teeth on the big ring and less teeth on the smallest ring. If you're really struggling on the hills, look for one with a "granny gear." Shimano markets them as "megarange." Basically, it's just a big gear that's a significant jump from the next biggest gear. It's not useful for much except for crawling up hills.

It's possible that your new cluster will be too big of a range for your current derailleur. It's hard to find precise specs on your bike, so I can't say for sure.You'll have to consult the derailleur manufacturer's technical manual to figure out whether your derailleur will accommodate the wider range of your new cluster. 

It's also possible that your chain won't be long enough. I'd probably just put the new rear cluster on and see if you can shift through all the gears. If you can, great. If you can't get up onto the bigger rings, you'll need a longer chain. Sheldon Brown once again has an article on how to find the right chain length for your gear setup. 

Even if you need a new chain and derailleur, I'm betting that you could get the whole rear done with comparable parts for well under $100 US, probably closer to $50. If you just need a cassette or freewheel, it'll probably be under $20.

If you're absolutely determined to get gears in the front, you'll definitely need a front derailleur, front shifter, and cables and housing. You'll also need additional chainrings. You can buy them individually if your crankset will accept them. However, it's probably cheaper to just buy a new crankset. You'll also likely need a longer chain, if you haven't already bought one for the rear. You may also need a new bottom bracket with a longer spindle, but maybe not. By the time you're finished with all of that, you're probably talking about at least $100 for just the front, maybe closer to $200. That's somewhere between a quarter and half of the cost of your bike. If you have to buy any tools or pay for labor, it gets even more expensive. If your bike's in decent condition, you're better off selling it and putting that money toward a new bike.