Тратить - is the literal translation (тратить деньги - to spend money) but speaking about time it has some association with 'to waste'. Well, not exatly waste. But it means 'so spend time as a valuable resource'.
I'm not sure if English 'spent' means the same. Does it?
Он потратил свою жизнь на...
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Проводить - what we make with our time _without_ this 'valuable resource' thing. Here the time is something, that will be filled with this or that occupation anyway.
Он провел свою жизнь, сочиняя романы.
Он провел свою жизнь за сочинением романов.
(I substituted writing with 'composing', as latter is more common in Russian for this situation)
When a noun exists on it's own, not just as a verbal noun, you can use preposition в:
Он проводит свою жизнь в поездках/разъездах/путешествиях/... на автобусе.
I don't know what 'travelling about' means:)