The Lost in Space Wiki has your answer here:
The spacecraft is propelled by atomic propulsion, using the substance
deutronium as a power source. It also appeared to have an ion drive,
and a magnetic drive. Small rockets came from unknown locations along
the rim of the ship.
This is also supported on the Wikipedia page
Specifically, the previously linked Lost in Space Wiki page mentions that
it has a radial reactor
In the very first episode (apologies - the video formerly there no longer exists) you will recall that we are told in a 'documentary' of the ship that the Jupiter 2 was powered by atomic motors, and the mining for deutronium is a regular reason used in the plot for why they don't leave whatever planet they've crashed onto.
In terms of lift-off, there is the use of an Anti-Gravity Drive (emphasis mine):
The anti-gravity drive system was designed for use in vehicle liftoff
and touch down. The anti-gravity drive consumes 250 megawatts at full
power and is capable of delivering up to ten g’s of acceleration. In
terms of the Earth’s gravitational field at sea level, this translates
to 55,000 pounds of thrust. ... Visible light is given off as a
by-product of each revolution of the generated field. The anti-gravity
drive is throttled back when the pull of gravity on the spacecraft is
less than 1/20th of Earth’s gravity at sea level.
Full-power launches are characterized by a loud hum of the antigravity
engines, and a glow around the ship caused by the magnetic fields
being generated at the time. Such a launch was used to leave Earth.
For 'flying around'
When they are actually in space, they use the Deutronium Annihilation Drive:
Deep space propulsion is accomplished with two deutronium-annihilation
atomic motors. Theoretically, these engines are capable of producing
unlimited thrust and speed. Photons are created through deutronium
annihilation in the hafnium carbide reactor chamber located in the
center of the lower region of the spacecraft. The photons radiate
through the urns projecting from the Thompson field projector. These
engines cannot be activated except in deep space. Operation within the
atmosphere of a planet would result in life-threatening contamination
due to dangerous radioactive exhaust.