When entering the UK for the first time, at a minimum, students are advised to bring:
Tier 4 (General) Student Visa:
Letter of Acceptance
English language test certificate (if you have taken a Secure English Language Test)
Short-term Study Visa:
Letter of Acceptance
English language test certificate (this is not a legal requirement but you may wish to bring one with you if you have one)
Flights to the UK via the Republic of Ireland
This information is for nationals of the USA, Japan, South Korea and other non-visa nationals who are planning to travel to the UK via the Republic of Ireland, and enter the UK as a short-term student.
As a non-visa national, you can indeed normally travel to the UK and apply when you arrive for immigration permission to enter as a short-term student (or other type of visitor) for up to six months. Non-visa national students on short courses often choose to do this rather than applying for entry clearance before travelling. However, if you arrive in the UK from elsewhere in the Common Travel Area (CTA, this includes the Republic of Ireland), you will not have the opportunity to do this because there is no immigration control at your UK arrival point. You should therefore apply for UK entry clearance (a visa) as a short-term student or Tier 4 student in your home country before travelling to the Republic of Ireland.
If, despite the above advice, you do enter the UK from the Republic of Ireland with no specific UK immigration permission, and you are a non-visa visa national, you will in most cases automatically have permission to be in the UK only for three months as a tourist, not for six months as a student. The three months may not be enough time to complete your short course. If you enter the UK from the CTA and need to be in the UK for more than the three months, you will need to leave the UK within the three month period, then re-enter asking for immigration permission to enter as a short-term student, for a maximum visit of six months.
Note also that any immigration permission you are given on arrival in the Republic of Ireland is for the Republic of Ireland, not for the UK. This is only an issue when arriving via the Republic of Ireland (for example, via Dublin). Northern Ireland is part of the UK, so if you fly directly to Northern Ireland (for example direct to Belfast), the problem does not arise.