Fortunately, many of the Gundams are easy to place in an in-universe chronological order, as the first entry in the franchise established the in-universe Universal Century (UC) calendar. Most of the shared-universe Gundams take place in the Universal Century chronolgy, and for those specific shows they always establish what year they take place in. This list is going to be a little ugly because SE doesn't support proper tables, but I'll be doing the in-universe UC year, then the year of release.
- 1979 Mobile Suit Gundam (the original series that started the franchise)
- 1989 Mobile Suit Gundam 0080: War in the Pocket
- 1996 Mobile Suit Gundam: The 08th MS Team
- 2004 Mobile Suit Gundam MS IGLOO: The Hidden One Year War
- 2006 Mobile Suit Gundam MS IGLOO: Apocalypse 0079
- 2009 Mobile Suit Gundam MS IGLOO 2: The Gravity Front
- 2015-(ongoing) Mobile Suit Gundam: The Origin
- 1991 Mobile Suit Gundam 0083: Stardust Memory
- 1985 Mobile Suit Zeta Gundam (some compilation movies were released in 2005)
- 2001 Gundam Neo Experience 0087: Green Divers
- 1986 Mobile Suit Gundam ZZ
- 1988 Mobile Suit Gundam: Char's Counterattack
- 2010 Mobile Suit Gundam Unicorn
- 1991 Mobile Suit Gundam F91
- 1993 Mobile Suit Victory Gundam
In terms of viewing order, most are pretty standalone. If you have the stomach for old-school anime, I'd recommend starting with the original Mobile Suit Gundam, as that will get you thoroughly familiar with key events in the universe. The year UC0079 is a busy time as there's a wide-spread war taking place, which is why so many series take place roughly simultaneously then. After watching the original, I'd recommend you read quick summaries of the other series, and try to watch ones that interest you. Char's Counterattack is a notable non-standalone story that directly comes from the people and events of the original Mobile Suit Gundam.
The rest of the series below take place in different universes than the above titles. Many are one-off series/OVAs. These are in no particular order. The header is the calendar system the series takes place in, which is another way the makers of Gundam make it clear these are in different universes.
(Yes, these take place in our real-world timeline)
- 2010 Model Suit Gunpla Builders Beginning G (I don't think this one is sharing much with the rest in this list, as it takes place 'in the present' in a world where Gundam is a fictional franchise)
- 2007 Mobile Suit Gundam 00
- 2010 Mobile Suit Gundam 00 the Movie: Awakening of the Trailblazer
- 1995 Mobile Suit Gundam Wing
- 1997 Gundam Wing: Endless Waltz
- AG115, AG140
- 2011-(ongoing) Mobile Suit Gundam AGE
- 1999 ∀ Gundam (pronounced Turn-A Gundam)
Gundam SEED was a recent extremely popular Gundam series. It pulls elements from past Gundams and shares some similarities as a result. I'd recommend this as an alternative to watching the original Mobile Suit Gundam.
- 2002 Mobile Suit Gundam SEED
- 2004 Mobile Suit Gundam SEED Destiny
- 2006 Mobile Suit Gundam SEED C.E. 73: Stargazer
- 1994 Mobile Fighter G Gundam
An alternative way to get into Gundam is to watch a few of these side universe shows. They share thematic similarities with the UC Gundams, but don't have as much history due to the lack of a shared continuity.
This list is not exhaustive, there are other series, and of course new Gundam works are still being produced. Most of this information I gathered from this Wikipedia list which also has links to plot summaries and more information on all these and more.