While "Hinduism" is actually not a single religion, but infact a collection of multiple faiths and philosophies, I would say the following are the common core beliefs shared by most adherents of Hindu religion irrespective of their philosophy, or the sect which they belong to.
1. The belief in the Param-atma or Supreme Soul (God)
2. The existence of Jiva-atma (individual souls). Every conscious life form has a soul.
3. All individual souls go through the cycle of birth, life and death over and over again (reincarnation)
4. The belief in Karma - Every thought, speech and action that a soul engages in accumulates and influences directly or indirectly the life experiences of the person in the current or future lives.
5. If a soul needs to be freed from the cycle of birth & death, then it needs be free of all ego and free of all karma. The soul needs to "realise" the Paramatma. This realisation of God, will free the soul from rebirth and the soul is then said to attain moksha.
While the above five are the common beliefs among all philosophical branches of Hinduism, where we can find a huge variety in beliefs is in:
1. The nature of the Paramatma - Some believe that God has no attributes at all, others believe in a personal God.
2. The nature of relationship between Paramatma and Jivaatma - Some believe that the individual soul is but a part of the super-soul and when it attains moksha, it joins the super-soul as a drop of water would join the ocean and becomes indistinguishable. Others believe that the distinction between the Paramatma and Jivaatma are always maintained.
3. The way to attain Moksha - There are many different ways to rid the self of ego - complete devotion to God, selfless service, meditation, etc.