This is a good video that explains in simple terms the Spot Rate, Forward rate and forward price from CFA Level 2 perspective.
Eg: yr1 spot rate = 3%, yr2 = 4% yr3 = 6%
Assuming FV = $1000 and a cash flow is $50 per year, use scientific calculator to compute PV of these cash flows.
Using the PV, the YTM can be obtained.
To obtain forward rates using these spot rates, eg. 1 year forward rate 1 year from now, we can use yr2 spot rate and yr1 spot rate to calculate the result. In other words, if I need to earn 4% overall in 2 years and I am investing at 3% for first year, then what should be my rate 1 year from now for the next 1 year, so I can take the money from year 1 investment and reinvest it at this new rate. Based on no arbitrate theory (1+ yr1 spotrate) * (1+ yr 1 forward rate for 1 yr) = (1 + yr2 spot rate)^2
Trying one or more examples will make this really simple and intuitive.