I take it that I can have one transmitter and link it to as many receivers/boats as I want?
Indeed you can. That's actually an area where the Radiolink scores if you want to cut costs, since the extra receivers are the cheapest on the market, at under £10.
You don't need to worry about Mode 1/2. What it refers to is more important for aircraft flying.
Speaking simply, you have 2 sticks on a radio, each providing 2 controls (up/down and left/right). So there are 4 things you can control in total. For a boat you usually use 2 (rudder and speed). For an aeroplane you use all 4 (rudder, elevator, aileron and speed). It's up to you which stick movement on the radio works which control on the model, but there are some obvious choices. For a boat, one of the left/right sticks might do steering, and one of the up/down sticks might do speed - it doesn't really matter which.
For an aeroplane, there are two possible 'obvious' choices - speed on the left stick or the right. In Europe modelers usually use one way - in the US they usually use the other way. That's Mode 1 or 2. For an aeroplane, if you let someone else fly it, it's critically important for them to know which controls do what, so standardisation of control inputs is very important. For a boat it's much less so.
In any case, most radios nowadays have a little switch that lets you change the controls between modes 1 and 2 very easily. The Radiolink and the Planet both have this.