• Firstly, it depends on your home state. If you belong to the northeastern region or states like Haryana ,Punjab, srinagar, Uttar Pradesh, then you have high probabilities of acquiring seats in NITS of your home state even in some of the core branches.
• If you belong to southern states or states like Jaipur, MP, Maharashtra you probably have least chances of getting any NIT.
• You can then approach to apply only private universities of your interest. Top private universities being Thapar, College of engineering -pune, RPeS-Ahmedabad, Deendayal Institute of petroleum engineering- Dehradun, Symbiosis Institute of technology-driven Pune.

As far as the fee is concerned, In NITs the annual fee accounts upto 1.25 lakhs per annum.  And various other private universities have wide varying fee higher than those of NITs.

(The query is answered assuming that you belong to general category.)