If I play a unit and use Twilight Step to move him to a battlefield where there are enemy units, can I move my units in base to join him or does a showdown begin before I can also make the rest of my units join battle?

No. After Twilight Step resolves and moves the unit, a Cleanup immediately stages a Combat at that battlefield and the Showdown begins — there is no window to then standard move additional units from your Base.

Here's why: The move from Twilight Step applies Contested to the enemy-occupied battlefield. Per Rule 448, after the move completes, a Cleanup runs. During that Cleanup, a Combat is staged (Rule 323.9), and as part of that same Cleanup (Rule 323.11) the Turn Player must initiate the Showdown. You never return to a Neutral Open State where a standard move would be legal — the state transitions directly from "spell resolving" to "Cleanup" to "Showdown."

This is consistent with the FAQ on similar effects: after using Showstopper or other single-unit move spells, you cannot standard move additional units before the showdown resolves (FAQ #1026, #2357). Standard moves are also flatly prohibited during a Showdown (Rule 144.1.c).

If you want multiple units in that fight, you have one clean option: use a Standard Move to move all your desired units (including the one you'd target with Twilight Step) simultaneously from your Base to the enemy battlefield as a single action (Rule 144.3). They all arrive together before any Showdown begins.