Honestly, I think you’ve answered your own question. If it works in Firefox, use Firefox.
Browsers are complex beasts, and websites can be complex beasts too. When you combine them, things get geometrically more complex. All three of the top browsers: Internet Explorer, Firefox, and Chrome, are good, solid browsers, as are many others like Safari and Opera.
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Pick a favorite, and then add another
I have no problem recommending that you use any one of them as your primary browsers. Pick whichever one feels best to you; the one whose interface you feel is the most intuitive; perhaps the one you feel performs the best on the sites that you typically visit. Seriously, any of them will do just fine these days.
Then install another one. If you use and like IE and then have either Firefox or Chrome installed as well, make sure that you leave the default browser set to the one you like the most. However, I can pretty much guarantee that sooner or later you’re going to want one of the others.
Some websites work well in every browser, and some don’t.
Compatibility is complex
Cross-browser compatibility is a difficult thing to achieve with 100% success. While there are standards, there are enough variations in how things work that it’s not that hard to accidentally overlook something that works a little differently in one browser as compared to another. The site that works fantastic in Firefox just might not look good in IE.
On one hand it’s the website’s fault for not doing it correctly. On the other hand, it’s unrealistic to expect every single website to test against every possible browser. Stuff happens.
Internet Explorer had a reputation for a long time for not being standards-compliant. Websites that were written correctly would simply not display correctly in IE.
And it was, in fact, Internet Explorer’s fault. Period. And yet, it was the most popular browser by virtue of being included in Windows.
IE’s non-compliant legacy
IE itself has gotten a lot better in recent versions. They even added what they called a “compatibility mode” for websites that don’t display well. But there are people that still swear that IE doesn’t display many websites properly. All I can say is, that could be.
The other problem though is that for a long time, many websites were actually written to work specifically with that non-standards-compliant Internet Explorer. What that means is that even though IE was doing things wrong, websites wanted to work with the most popular browser, and thus they did things wrong as well in order to work with IE.
As IE, over time, did a better job of adhering to standards, now those old websites that were written to work with old non-compliant IE won’t work properly in other browsers or even later versions of IE.
The bottom line here is that you can see pretty quickly, it’s a bit of a mess. The practical, pragmatic solution for you as an average user is very, very simple. If the browser you use is not doing what you need it to, try a different one. It costs nothing to go back and forth.