Adding front derailleur vs upgrading the rear one on a (ladies) bicycle
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.