Welcome, White Raven!
It sounds to me like what you’ve got is the Milton Bradley version of Axis & Allies, also known as “Classic”. It was indeed first published in 1984. There were three editions of this version, with the 2nd Edition (1986) being the most popular. As a result of this, many people also call the game “2nd Edition”. The forum for this version can be found here.
Starting in 2004, there have been several new versions of Axis & Allies published by Avalon Hill. The most current of these are the 1942 and 1940 Editions. The 1942 Edition is the “base” game that is the modern equivalent of what you have. We’re currently in the 1942 Edition forum. The 1940 Edition is a much larger and more detailed game, which actually consists of two Theatre games (Europe 1940 and Pacific 1940) which can be combined into a Global game.
There isn’t really any way to “upgrade” the game you have to a newer version, as the boards are different and there are new unit types. Of course, you could create a “hybrid” version by adding some of the new rules and picking up pieces to use for the new units. However, just buying a 1942 Edition is inexpensive enough that it’s not really worth it to try to do this.
Of course, there is no reason why you can’t continue to play the game that you have. It’s still a fine game, and many people continue to enjoy it. (There’s a reason it’s called “Classic”.) However, if you want to move up to a more modern version, I invite you to check out the Avalon Hill website for more information on the newer versions. In many cases, the rules are available there for download.
Feel free to ask more questions here as well. There are many friendly folks willing to answer them.