Buddhism view regarding happiness and contentment

A comparison of sensual pleasures with other forms of happiness:

[...] The pleasure and joy arising dependent on these five strands of sense desire, that is called sensual pleasure.

"Now, if someone were to say: 'This is the highest pleasure and joy that can be experienced,' I would not concede that. And why not? Because there is another kind of pleasure which surpasses that pleasure and is more sublime. And what is this pleasure? Here, quite secluded from sensual desires, secluded from unwholesome states of mind, a monk enters upon and abides in the first meditative absorption (jhana), which is accompanied by thought conception and discursive thinking and has in it joy and pleasure born of seclusion. This is the other kind of pleasure which surpasses that (sense) pleasure and is more sublime.

"If someone were to say: 'This is the highest pleasure that can be experienced,' I would not concede that. And why not? Because there is another kind of pleasure which surpasses that pleasure and is more sublime. And what is that pleasure? Here, with the stilling of thought conception and discursive thinking... a monk enters upon and abides in the second meditative absorption... in the sphere of the infinity of space... of the infinity of consciousness... of no-thingness... of neither-perception-nor-non-perception. [...] the cessation of perception and feeling. [...] This is the other kind of pleasure which surpasses that pleasure and is more sublime.

"It may happen, Ananda, that Wanderers of other sects will be saying this: 'The recluse Gotama speaks of the Cessation of Perception and Feeling and describes it as pleasure. What is this (pleasure) and how is this (a pleasure)?'

"Those who say so, should be told: 'The Blessed One describes as pleasure not only the feeling of pleasure. But a Tathagata describes as pleasure whenever and whereinsoever it is obtained.'"

-- Pañcakanga Sutta, SN 36.19

In another exposition, the Buddha distinguishes between worldly joys and unworldly joys. For example:

"Now, O monks, what is worldly joy? There are these five cords of sense desire [...]. It is the joy that arises dependent on these five cords of sense desire which is called 'worldly joy.

"Now what is unworldly joy? Quite secluded from sense desires, secluded from unwholesome states of mind, a monk enters upon and abides in the first meditative absorption [...]. This is called 'unworldly joy.

-- Niramisa Sutta: Unworldly, SN 36.31

It can be seen in the suttas that, in general, happiness, joy, contentment (or, specifically, the wholesome kinds) are at the very heart of the teachings, where nibbāna is declared to be the foremost happiness. On that note:

Bhikkhus, it is impossible that a bhikkhu who considers nibbāna to be suffering will possess a conviction in conformity [with the teaching]. [...] [But] it is possible that a bhikkhu who considers nibbāna to be hapiness will possess a conviction in conformity [with the teaching].

-- Nibbāna [Bodhi Trans.], AN 6:101

Also, as @yuttadhammo explained in his answer, in the context of samatha:

When the mind is directed to some satisfactory image, happiness is born. From this happiness, joy is then born. With a joyful mind, the body relaxes. A relaxed body feels content, and the mind of one content becomes concentrated.

-- SN 47.10

We also see qualities like rapture (piti) in the seven factors of enlightenment. Finally, in a dependent arising formulation of the path (see Ven. Bodhi's exposition on this particular sutta):

Emancipation <- Dispassion <- Disenchantment <- Knowledge-and-vision-of-things-as-they-are <- Concentration <- Happiness (Sukha) <- Tranquillity <- Rapture (Pīti) <- Joy (Pāmujja) <- Faith <- Suffering <- Birth <- Existence <- Clinging <- Craving <- Feeling <- Contact <- the Six Sense-Bases <- Name-and-Form <- Consciousness <- (kamma-) formations <- Ignorance

-- Upanisa Sutta, SN 12.23

This sort of happiness is certainly recognized by the Buddha; it is encouraged in many places as being conducive to concentration, e.g.:

2.“I thought: ‘Why am I afraid of that pleasure that has nothing to do with sensual pleasures and unwholesome states?’ I thought: ‘I am not afraid of that pleasure since it has nothing to do with sensual pleasures and unwholesome states.’

-- MN 36 (Bodhi, trans)

In final analysis, however, it must be discarded as useless - unstable, unsatisfying and uncontrollable:

(i) “And what, bhikkhus, is the gratification in the case of feelings? Here, bhikkhus, quite secluded from sensual pleasures, secluded from unwholesome states, a bhikkhu enters upon and abides in the first jhāna, which is accompanied by applied and sustained thought, with rapture and pleasure born of seclusion. On such an occasion he does not choose for his own affliction, or for another’s affliction, or for the affliction of both. On that occasion he feels only feeling that is free from affliction. The highest gratification in the case of feelings is freedom from affliction, I say.

33–35.“Again, with the stilling of applied and sustained thought, a bhikkhu enters upon and abides in the second jhāna…With the fading away as well of rapture…he enters upon and abides in the third jhāna…With the abandoning of pleasure and pain he enters upon and abides in the fourth jhāna…On such an occasion he does not choose for his own affliction, or for another’s affliction, or for the affliction of both. On that occasion he feels only feeling that is free from affliction. The highest gratification in the case of feelings is freedom from affliction, I say.

36.(ii) “And what, bhikkhus, is the danger in the case of feelings? Feelings are impermanent, suffering, and subject to change. This is the danger in the case of feelings.

37.(iii) “And what, bhikkhus, is the escape in the case of feelings? It is the removal of desire and lust, the abandonment of desire and lust for feelings. This is the escape in the case of feelings.

-- MN 13 (Bodhi, trans)