Starting with TOS is fine. But if you want you can also watch Star Trek Enterprise first. Enterprise is a prequel and takes place before TOS in the 22nd century. It's the first warp 5 ship, and it's called Enterprise. It's strictly a Starfleet ship, as in this point in time the United Federation of Planets has not happened yet. It's an awesome series. Particularly in the 3rd and 4th season.
I'll give you a break down of the complete chronological sequence of the Star Trek Universe.
Star Trek Enterprise
Season 1
Season 2
Season 3
Season 4
Star Trek The Original Series
Season 1
Season 2
Season 3
Star Trek The Animated Series
Season 1
Season 2
Star Trek The Motion Picture
Star Trek II The Wrath of Kahn
Star Trek III The Search For Spock
Star Trek IV The Voyage Home
Star Trek V The Final Frontier
Star Trek VI The Undiscovered Country
Star Trek The Next Generation
Season 1
Season 2
Season 3
Season 4
Season 5
now it gets tricky, Star Trek Deep Space Nine runs side by side with Next Generation in the exact same time period. I always reccomend that people watch Next Generation and Deep Space Nine together at this point.
TNG Season 6/DS9 Season 1. However Season 1 of DS9 begins part way through Season 6 of TNG and that's why Season 1 is shorter and only has 19 episodes instead of the usual 26. Begin DS9 after watching TNG Season 6 Chain of Command Part 2.
TNG Season 7/DS9 Season 2. These Seasons you can watch together from the start. They started together and ran together from the start.
Deep Space Nine Season 3/Star Trek Generations/Star Trek Voyager Season 1. Generations is the 7th movie and is the first of four TNG movies. Watch it after DS9's 3rd season episode Meridian. Now half way through Season 3, Star Trek Voyager begins. It's first season is also short, shorter than DS9's first season was. It has 15 episodes.
So about half way through, start Voyager season 1, and finish out DS9 season 3.
Deep Space Nine Season 4/Voyager Season 2
Deep Space Nine Season 5/Voyager Season 3/Star Trek First Contact. First Contact is the 8th Star Trek film and is the 2nd of 4 TNG movies. Watch it around DS9's 5th Season episode Rapture.
Deep Space Nine Season 6/Voyager Season 4
Deep Space Nine Season 7/Voyager Season 5/Star Trek Insurrection. Insurrection is the 9th Star Trek movie and is 3rd of 4 TNG movies. Watch it around DS9's 7th Season episode It's Only a Paper Moon.
Voyager Season 6
Voyager Season 7
Star Trek Nemesis. Nemesis is the 10th and final TNG film, and the final film of the Roddenberry/Berman Trek era.
Star Trek 2009, is a new reboot/remake of TOS. It takes place in an alternate timeline where anything new or different can happen.
Star Trek Into Darkness comes out May 17th this year as a sequel to Star Trek 2009.
This is my reccomendation of the sequence in which you view everything.
The years that everything was produced is as follows.
TOS: 1966-1969 3 Seasons
TAS: 1973-1974 2 Seasons
The Motion Picture: 1979
Wrath of Kahn: 1982
Search For Spock: 1984
The Voyage Home: 1986
TNG: 1987-1994 7 Seasons
The Final Frontier: 1989
The Undiscovered Country: 1991
Deep Space Nine: 1993-1999 7 Seasons
Voyager: 1995-2001 7 Seasons
Generations: 1994
First Contact: 1996
Insurrection: 1998
Nemesis 2002
Enterprise: 2001-2005 4 Seasons