Moving to slovakia from UK

Have a look here, they have all the info http://mic.iom.sk/en/ . The website does seem to be down for the moment.
Residence of Union citizens and their family members   
A citizen of the Union is entitled to a residence in the territory of the Slovak Republic during a period longer than three months, if he/she:
is employed;
is a self-employed person;
studies in primary or secondary school or at a university;
has sufficient financial means for himself as well as for his family members, to prevent them from burdening the social security system during their stay  and has health insurance in Slovakia;
will potentially find employment;
is a family member of a Union citizen and is accompanying or joining him and meets the above mentioned condition.
Registration of permanent residence in the territory of the Slovak Republic

A Union citizen staying in Slovakia for a period longer than three months is obliged to apply for registration of residence within 30 days from entering the territory of the Slovak Republic. The application for a free of charge registration is submitted in person on an official form at the competent Alien Police Department together with a valid identification card or a valid travel document and a document confirming one of the matters above (employment contract, Trade Licence, confirmation from school or a bank account statement).

On the day of submitting a complete application, the Alien Police Department will issue a confirmation about the residence registration of a Union citizen, which will state the name, surname and the address of the registered person as well as the date of registration. If the Union citizen fails to submit a document on accommodation, the municipality where he/she will be staying will be stated as the address.

Union Citizens may apply for issuance of a residence document valid for 5 years (plastic card) in person at the competent Alien Police Department according to the place of residence in Slovakia. Contact details of Alien Police departments can be found on this web site.  A valid travel document or valid identification card, 2 photographs (3x3.5 cm) and a document on accommodation need to be attached to the application for issuance of a residence document.

For registration of residence as well as for issuance of a residence document, Union citizens must use the form available here.

The right of  a Union citizen to permanent residence

A Union citizen is entitled to a permanent residence, if he has legally resided in Slovakia for an uninterrupted period of five years. In specific cases, it is possible to apply for permanent residence even earlier than after five years of uninterrupted residence.

Obligation of the employer

If the Union citizen comes to Slovakia for the purpose of employment, the employer is obliged to notify the local Office of Labour, Social Affairs and Family within 7 working days from the start of the employment by sending a so-called Information Card (see the EURES web site).

Termination of the right to residence of a Union citizen

The entitlement to residence or the entitlement to permanent residence of a Union citizen expires, if:
the person notifies the Alien Police Department in writing of the residence termination;
the person was deported;
the Alien Police Department revoked his right to residence or to permanent residence;
the person died or was declared dead;
the person acquired citizenship of the Slovak Republic.
Notice of the stay

A Union citizen is also obliged to notify the Alien Police Department of the start as well as of his/her place of residence within 10 working days from the day of entering Slovakia, unless the provider of accommodation (e.g. hotel) does it instead. This means the notification is necessary in case the Union citizen lives with relatives or friends.

It is necessary to submit a confirmation of accommodation containing the name of the person hosting the Union citizen including the person's ID number.

More information can be found on the web site of the Central Public Administration portal, on the web site of the Bureau of Alien and Border Police and on the web site of the EURES portal.

No special paperwork is needed from the UK however at the end of the first tax year in Slovakia you will need some of your UK tax documents to show if you worked during the financial year of 2013 within the EU.  I found that the documentation did not need to be translated and was processed quickly by the tax office while I waited.
The post about the residence card is important as the ID card is used for almost everything and is something to keep with you at all time.
If you are bringing a UK car you will need valid car insurance and can only keep it here for a certain length of time.
Good luck and let me know if you need any specific information.