Is it normal to have veins like white line with small bumps/pimple attached with it under the blue vein underneath the tongue?

Yes: Your description sounds like a normal variant of the tongue and surrounding structures.

Normal: Was you are describing is normal anatomy. You are looking at blood vessels and salivary gland ducts. Your Dentist will routinely examine the floor of your mouth (as well as the rest of your oral structures) at your regular check-ups. If it has been more than 6 months, your overdue.