What is the ultimate goal of Buddhism? Is it nirvana?

That's easy, there is a sutta just for you

https://www.accesstoinsight.org/tipitaka/an/an11/an11.001.than.html

and the '''noble search'' https://www.accesstoinsight.org/tipitaka/mn/mn.026.than.html

the goal is to ''understand as it actually is.'' which means understanding the origin of dukkha, dukkha itself and the cessation of dukkha http://www.buddha-vacana.org/sutta/samyutta/maha/sn56-001.html

It turns out that understanding as it really is means understanding plenty of things. https://www.accesstoinsight.org/tipitaka/an/an06/an06.063.than.html

The ultimate goal in Buddhism is the end of suffering. But why would you want to reach the end of suffering? The reason is to attain the only permanent happiness, which is Nirvana. So, one's mission in life in Buddhism is really the pursuit of happiness.

From Dhammapada 203-204:

203. Hunger is the worst disease, conditioned things the worst suffering. Knowing this as it really is, the wise realize Nibbana, the highest bliss.

204. Health is the most precious gain and contentment the greatest wealth. A trustworthy person is the best kinsman, Nibbana the highest bliss.

The Buddha discovered that both over-indulgence and over-asceticism are not conducive to the path to the end of suffering. So, he prescribed the middle way through the Noble Eightfold Path.

If you follow the middle way, you can have long term, medium term and short term goals of happiness.

The long term goal associated with this is attaining Nirvana (permanent happiness).

The medium term goal would be to try to achieve: at least stream entry for Theravada and strong cultivation of Bodhicitta for Mahayana. You can also get more info on stream entry in this YouTube video talk.

The short term goal would be to try to achieve and maintain "materialistic" happiness in this life and future lives (which includes avoiding unfortunate rebirths). At the very minimum, you need to keep the five precepts with heedfulness (appamada). Going a little further, you need to practise more of virtue (sila) with heedfulness (appamada).

This short term goal is described in the Ittha Sutta:

Long life, beauty, status, honor, heaven, high birth: To those who delight in aspiring for these things in great measure, continuously, the wise praise heedfulness in making merit.

The wise person, heedful, acquires a two-fold welfare: welfare in this life & welfare in the next. By breaking through to his welfare he's called prudent, wise.

For lay followers, there is plenty of advice on achieving the short to medium term goals in the Gihi Sutta (or Discourse to the Householder), Sigalovada Sutta, Dighajanu Sutta and Anana Sutta. The minimum training rules imposed on lay followers are the five precepts. Also see this answer for the question "Can a Buddhist own and run a billion dollar business?"

For those in a hurry to Nirvana, there are the more advanced training paths of anagarika (sort of a pre-monk or pre-nun), novice monk or nun, and fully ordained monk or nun. Please see this answer for details.

The Buddha's revelation of Nirvana is officially defined as the cessation of greed, hatred & delusion or 'The Deathless'. For simple minded people, the Dhammapada explains 'Nirvana is the highest happiness'. The Buddha's revelation of Nirvana is the 1st systematically explained historical revelation of liberation. The doctrines of liberation of other modern religions logically come from Buddhism. Therefore, it is improper to saying Nirvana doesn't seem to be a great goal. The Buddha taught there can only be one Buddha in a world system. The Buddha is the saviour or light of the whole world. The 'Eternal Life' of Jesus is logically 'The Deathless' taught by the Buddha.

"Whoever discovers the interpretation of these sayings will not taste death."

"Those who seek should not stop seeking until they find. When they find, they will be disturbed. When they are disturbed, they will marvel, and will reign over all. [And after they have reigned they will rest.]"

"If your leaders say to you, 'Look, the (Father's) kingdom is in the sky,' then the birds of the sky will precede you. If they say to you, 'It is in the sea,' then the fish will precede you. Rather, the (Father's) kingdom is within you and it is outside you."

The Gospel of Thomas

But what is the ultimate goal of Buddhism?

To begin with maybe it's not entirely about, perhaps even not at all about, an "ultimate goal".

Just as the great ocean, monks, gradually inclines, gradually slopes, gradually slants, certainly does not fall away abruptly, so, monks, in this Dhamma and Discipline there is a gradual training, a gradual performance, a gradual practice, it certainly does not have an abrupt penetration of knowledge. That, monks, in this Dhamma and Discipline there is a gradual training, a gradual performance, a gradual practice, and it certainly does not have an abrupt penetration of knowledge, is the first wonderful and marvellous thing, monks, about this Dhamma and Discipline, which, having seen and considered, the monks delight in this Dhamma and Discipline.

from Ud 5.5

Perhaps it begins with doctrines of morality, harmlessness, kindness, prudence, duty -- for well-earned (albeit perhaps temporary) bliss, for the sake of others ... and an "absence of remorse" (which should result from "skilful virtue", and which doctrinally is one of the first steps towards towards proper joy and so on).

The goal to achieve nirvana is basically dying and never coming back, because Buddhism believes in rebirth and the cycle of samsara, which it believes is filled with sorrow and suffering.

That's kind of debatable, I guess. One venerable, on this site, wrote earlier,

I've talked about the subject before myself, and the conclusion I make is that it's not that Buddhists believe in rebirth, it's that we don't believe in death

Instead of saying "dying and never coming back", perhaps you might just as well say "living and never going away", eh?

One of the Buddha's titles the Tathagata and I'm not sure whether that means "thus-come" or "thus-gone" (someone said its etymology is ambiguous -- i.e. gata or agata).

And then there's e.g. this story from another Buddhist tradition (i.e. Zen), i.e.

If you think you really come and go, that is your delusion. Let me show you the path on which there is no coming and no going.

the cycle of samsara, which it believes is filled with sorrow and suffering

I'm not sure. Some schools teach that "suffering" is impermanent too -- that some "beings" suffer.

See also Did the Buddha really say that "life is suffering"?

one should try to achieve nirvana. Am I correct??

I'm not sure even of that: one "should"? Perhaps one "will"? Or that one "has", even, the Buddha himself for example?

I like the "four noble truths" i.e. which say that perpetually "craving" for what you can't have, craving to keep what you can't keep, etc., is a cause of "suffering" -- and that suffering and craving are both subject to cessation.

Some people don't much like that message though (or the way in which someone might try to explain it), see e.g. How to explain what Buddhism is? -- see also Did the Buddha teach the four noble truths to laypeople?