Today shares worth nothing, but startup may accept funding and that may raise valuation.
My concern is whether I will be liable to pay taxes on vested shares each year even though it's not a publicly traded company and valuation can be calculated in many ways.
Speaking for US taxes - it depends on a lot of things. The first of which are they NSO or ISO's? Most likely they are ISO's.
If they are ISO's, them vesting has no tax implications until you exercise them. The company should have given you an exercise price. This is the per option price you would pay to own the shares after the vesting date. Let's say they gave you 1,000 options each year for 3 years at an exercise price of $1.00. Then after that first year it would cost you $1,000 to exercise those options. You would then be liable for AMT gains from the current fair market value of the options minus the exercise price. Let's say that when you exercises the 1,000 options, they were worth $2.50. Then you would have had a "gain" of ( 2.50 - 1.00 )* 1000 = $1,500 even though you haven't sold them or received any money. This gain is used to compute your AMT. The AMT has certain allowances so depending on how much that "gain" is and your salary you may or may not owe extra taxes on the gain for the current year.
If this was confusing - do some googling on ISO AMT and then contact a lawyer.
A smaller question is "is the company really a startup?" The granting of shares to employees rather than options is more typical for companies organized to produce cash flows to share holders rather than spending available cash in pursuit of growth.
Good luck.