To make your Joomla upgrade process as smooth as possible, there are few things which have to be planed before upgrading your Joomla website
Design and Layout
First thing which you will have to consider before upgrading your Joomla is the design and layout of your current website. If you are using commercially available Joomla template For Example: Rocket Themes, JoomlaArt, then you will have to check with the respective template provider if they have same Joomla Template available for the new Joomla version or not. There are good chances that the template is available, but can have little different color scheme and backend interface. If the template is not available or is custom made and you want to keep the design same, you will have to re-create Joomla template for this new version of Joomla.
Next thing which you should check are the extensions availability. Many Joomla components are free and are no longer available or not updated. Its not recommended to use component which is not updated and bugs are not fixed regularly as this can have major security risk of getting your new Joomla website hacked. If the component is not available, you will have to search for similar component which provides similar functionality.
Data and Content
It might happen that you have articles and content pages which are no longer relevant and of no use in today’s time. It’s better to get rid of such articles before you dive into Joomla upgrade. Do remember to 301 redirect the missing pages/url to your home page.
Many times there are unused extensions (components, modules and plugins) which can complicate the upgrade process. It’s recommended that such extensions should be uninstalled first. Better will be just disable them as some times uninstalling can get you in trouble and website can go down.
Even after detailed planning things can go wrong. So the next most important part is to have working backup of your current Joomla website so that if something goes wrong you can roll back.
Hosting and PHP/MySQL Versions
There are some minimum technical requirement which has to be satisfied for latest version of Joomla 3.4 You can see the requirement here. If your current hosting do not support latest php/mysql version this can create problem. Try to get in touch with hosting or consider moving to new hosting company will be better solution.