What should be my preference for the following colleges if I have no specific branch in mind: new IITs, old IITs, BITS Pilani, BITS Goa, BITS Hyderabad, BIT Mesra, BIT Sindri, local NIT, VIT, Shibpur,

It is difficult to give an appropriate solution if you are not opting for a specific branch. All the colleges you have mentioned are well-reputed institutes in India that offer quality education and placement support to their students. The comparison can only be made based on the branches. Some institutes, like IITs, are good at CSE, while others have efficient departments in mechanical or Electrical Engineering. The ranking and placement vary every year.

If you get a chance to pick BITS Goa in mechanical engineering and BITS Pilani in Civil Engineering, then opt for Mechanical engineering, which opens up more opportunities and placement support than civil engineering.