Don't get confused by that. The new Doctor Who series that started in 2005 stands fairly well on its own. Yes, they are all part of the same universe of the show that started in 1963, but I'm pretty sure you can just enjoy the new series without knowledge of the old one (1963-1989). Hard for me to say since I've been watching since 1978.
In terms of marketing, the old series is released in no particular order. Some will
never get released as they don't exist anymore. Some are released in special packs (like the
E-Space Trilogy with 112,113,114), consecutive related episodes or those grouped by themes/monsters (like
Beneath the Surface, with 52,62,131).
I think this is about right (do they count
Shada? I guess they count
Shada)
First Doctor (William Hartnell) -
Stories 1-29 - 1963 to 1966
Second Doctor (Patrick Troughton) -
Stories 30-50 - 1966 to 1969
Third Doctor (Jon Pertwee) -
Stories 51-74 - 1970 to 1974
Fourth Doctor (Tom Baker) -
Stories 75-116 - 1974 to 1981
Fifth Doctor (Peter Davison) -
Stories 117-136 - 1982 to 1984
Sixth Doctor (Colin Baker) -
Stories 137-147 - 1984 to 1986
Seventh Doctor (Sylvester McCoy) -
Stories 148-159 - 1987 to 1989
(break of 7 years)
Eighth Doctor (Paul McGann) -
Story 160 (TV Movie) - 1996*
(break of 9 years)
Here begins the "new" series that you're watching. They do not use the Story #s on the DVD releases, and instead refer to them as Series One, Two, etc. (as opposed to the older ones, "Season One, Season Two, etc.)
Ninth Doctor (Christopher Eccleston) -
Stories 161-173 "Series One" - 2005
Tenth Doctor (David Tennant) -
Stories 174-220 "Series Two, Three, Four and Year of Specials)" - 2006 to present
Eleventh Doctor (Matt Smith) - "Series Five" starts next year **
Along with (or after) the new ones you're watching, I suggest these classics:
#1 The Unearthly Child
#6 The Aztecs
#37 Tomb of the Cybermen
#46 The Invasion
#62 The Sea Devils
#65 The Three Doctors
#70 The Time Warrior
#76 The Ark in Space
#82 Pyramids of Mars
#91 Talons of Weng-Chiang
#115 The Keeper of Traken
#116 Logopolis
#117 Castrolvalva
#122 Earthshock
#136 The Caves of Androzani
#139 Vengeance on Varos
#141 The Two Doctors
#152 Remembrance of the Daleks
#158 The Curse of Fenric
The ones in bold are one from each Doctor 1-7* that IMO only are the best of those Doctors available.
* Doctor 8's TV Movie may never be released in the US for rights issues.
** EDIT - Just found out that this year of specials is not considered "Series Five". Series Five starts with Matt Smith next year.