You can tell Solve what variables you want to solve for:
sol = First@Solve[{F1[0] == Fi1, F1[1] == Fi2, F2[1] == Fi2,
F1'[1] == F2'[1]}, {k1, k2, k3, k4, k5, k6, k7, k8}]
You cannot plot the solutions directly, because some constants are still symbolic:
{F1[t], F2[t]} /. sol
You can assign all the remaining parameters the value 1, for example, and then plot the functions:
{F1[t], F2[t]} /. sol /.
Thread[{Fi1, Fi2, k1, k2, k3, k4, k5, k6, k7, k8} -> 1]
Plot[%, {t, 0, 2}]