
Lust haben auf
to feel like
The feeling of wanting to do or have something specific; an inclination or wish.
Ich habe heute Abend Lust auf Pizza.
I feel like pizza tonight.
desire / pleasure
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to feel like
The feeling of wanting to do or have something specific; an inclination or wish.
Ich habe heute Abend Lust auf Pizza.
I feel like pizza tonight.

pleasure
A feeling of joy, pleasure, or enthusiasm.
Er erledigte die Gartenarbeit mit großer Lust und Freude.
He did the gardening with great pleasure and joy.

as one pleases
In whatever way one wants at the moment; arbitrarily and spontaneously.
Im Urlaub kann man den Tag nach Lust und Laune gestalten.
On vacation, you can plan your day as you please.

lust
Strong sexual desire or craving.
In dem Roman werden Gefühle von Liebe und Lust thematisiert.
The novel explores feelings of love and lust.
Careful: 'Die Lust' usually just means 'Lust auf etwas haben' (to feel like something). It is much more general than the English word 'lust', which almost always refers to strong sexual desire.
Hast du Lust, heute Abend ins Kino zu gehen?
Do you feel like going to the movies tonight?
To express what someone feels like, you use the fixed construction 'Lust haben auf' (to feel like) followed by the accusative. This is one of the most common uses of the word.
Die Kinder haben Lust auf ein Eis.
The children feel like having an ice cream.
To express that you do not want to do something, the phrase 'keine Lust haben' (to not feel like it) is very common. Here you use 'keine' and not 'nicht'.
Ich habe heute keine Lust, zu kochen.
I don't feel like cooking today.
The idiom 'nach Lust und Laune' (as one pleases) means that you do something spontaneously and exactly the way you want at the moment, without a fixed plan.
Am Wochenende kann ich endlich ausschlafen und den Tag nach Lust und Laune gestalten.
On the weekend, I can finally sleep in and plan the day as I please.
joy
Having “Freude” in an activity means that you enjoy doing it and experience positive pleasure while doing so. “Lust” auf an activity refers more to the current desire or motivation to do it. So you can enjoy something over the long term but not feel like doing it right now.

Lea hat heute Lust zu malen und holt sofort ihre Farben.
Lea feels like painting today and immediately gets out her paints.
Lea hat große Freude am Malen und besucht jede Woche einen Malkurs.
Lea gets a great deal of joy from painting and attends a painting class every week.