ADC forwarding directly to UART
You will want to sample at a regular interval with as little jitter as possible so setup your MCU program to operate with a timer interrupt at your needed sample rate. Then at each interrupt read the results of the previous conversion and trigger the next conversion. Some MCUs have built in hardware than can be configured to trigger A/D conversions from a timer overflow. 

From the questions you have asked above it seems that you will need to do some investigation into your particular MCU to figure out how to use the timers and interrupts.

UARTs are good for 8-bit data bytes. The way you transmit 12-bits from an A/D converter is one of several ways. 

a) Send first 8-bits of the sample in one UART byte followed by 4-bits in the next UART byte.

b) Send the first 6-bits in a UART byte followed by the remaining 6-bits in the next UART byte. 

c) Both above approaches waste UART bandwidth on bit times not transmitting useful data so another scheme is to send two samples of the A/D (2 x 12 bits) packed into three UART bytes (3 x 8 bits). The packing can be sliced many ways but one common method used is to send the following sequence [Sample1(7:0)] [Sample2(7:0)] [Sample1(11:8)Sample2(11:8)].