I am using Version 10.6.8, have had this mac for awhile. I think I was recommended not to update the operating system as it would affect my software and became cautious of updating the browser also.
Macintosh OS X 10.6.8 (Snow Leopard) was the very last version of the Mac OS that was backward compatible with software written for the older PowerPC-based Macs. So, if you upgrade the OS on your Mac, any software that you have that was written for PowerPC will cease to operate and have to be replaced.
I, personally, have an old iMac in my office running OS X 10.6.8, specifically to run a couple of legacy apps that never made it past 10.6.8, and which I find invaluable.
So, it's true that if you update past 10.6.8 you are likely to lose some software and you are likely to have to pay to replace some of that software. If that software is Adobe software, it will likely cost a LOT of money to replace. (There are pretty nice substitutes for Adobe software that are much more reasonably priced, such as the products from Affinity, but they aren't worth talking about right now unless you decide to move to a MUCH more modern Mac and Mac OS.)
With regard to your Web browser, the news is a bit better. First, you are in no danger of losing anything if you upgrade to a better browser. You might not be able to easily transfer over your bookmarks from the old browser, but if you had to you could do it manually. You could easily run both browsers simultaneously to facilitate that.
The old version of Safari, which came with your Mac, is no longer stable, it's badly outdated, and it can't be updated because the version of Safari you could run back then was tied to the version of the OS you were using.
You can't update to a browser for OS X 10.6 that is perfectly modern, but you can get pretty close.
Personally I've switched to Firefox version 48.0.2 (an older, still available version) as the default browser on my old iMac running OS X 10.6.8. You can download that older version of Firefox for free from here:
Directory Listing: /pub/firefox/releases/48.0.2/
This version is quite modern. It's not 100% like a modern browser, but for the overwhelming number of sites that you might visit, it will do a good job.