Note: This answer is spoiler free. The links have been included for reference, but opening them likely will lead to spoilers.
From the Universal Soldier wiki:
After Universal Soldier was released, Carolco, the production company
that backed the film, went bankrupt. In 1995, Carolco sold the rights
of the series to Skyvision Entertainment in Toronto.
Two TV films,
Universal Soldier II: Brothers in Arms and Universal Soldier III:
Unfinished Business, were released direct-to-video...They were produced for Showtime / The Movie Channel as a
miniseries meant as a backdoor pilot for a series. Having almost no
input whatsoever from the original filmmakers, their canonical status
Soldier: The Return...has since been removed from the series
canon. An overwhelming critical and financial failure, The Return
contradicted several elements of the previous film's plot...
film in the series, Universal Soldier: Regeneration, disregards the
events of The Return entirely.
If I go to the Universal Soldier: The Return page, it says:
Subsequent films in the series ignore the events of The Return and
contradict it in some places throughout the series; today it is no
longer considered part of the series' canon.
Similarly, if I go directly to the Universal Soldier: Regeneration wiki page, it says:
The film is direct sequel to the original Universal Soldier from 1992,
unrelated to the two Universal Soldier television sequels that were
produced in 1998 and completely ignores the events from the 1999
theatrical sequel Universal Soldier: The Return.
Finally, the Universal Soldier: Day of Reckoning page says:
It is a semi-sequel to Universal Soldier: Regeneration.
So putting all of that together, it appears that the first film was released. Then two more direct to TV films were released. Then a fourth theatrical film was released, and has been removed from canon. A fifth film was then released, which served as a sequel to the first film, and a sixth film was released that serves as a semi-sequel to the fiffth film.
So, without the direct-to-tv films, I'd suggest this order:
- Universal Soldier
- Universal Soldier: Regeneration
- Universal Soldier: Day of Reckoning
You can then watch the two direct to TV films, and Universal Soldier: The Return, if desired.