You can simply replace the old TV with the new one directly.
Some TVs might be able to 'auto-sense' the presence of the Sky box but it's more likely that you'll need to set the correct source. To do so you usually just have to find a 'Source' button on the remote control (which might be labelled something like 'Src' or 'Input' but is more commonly indicated by a rounded rectangle with an arrow pointing into it). Pressing that will bring up a menu, where you use the 'Up' and 'Down' buttons to select 'HDMI' (or, more likely, because most modern sets have at least two HDMI inputs, 'HDMI 1' or 'HDMI 2'). Once you've found the correct source, press 'OK'(or 'Enter') and you should be fully set up.