Before answering specific question, you are liable to pay tax as per your bracket on the income generated.
I work with my partner and currently we transfer all earning on my personal bank account. Can this create any issue for me?
If you are paying your partner from your account, you would need to maintain proper paperwork to show the portion of money transferred is not income to you. Alternatively create a join Current Account. Move funds there and then move it to your respective accounts.
Which sort off account should be talk and by whose name?
Can be any account [Savings/Current]. If you are doing more withdrawls open Current else open Savings. It does not matter on whos name the account is. Paperwork to show income matters from tax point of view.
What should we take care while transfering money from freelance site to bank?
Is there any other alternative to bank?
There is paypal etc. However ultimately it flows into a Bank Account.
What are other things to be kept in mind?
Keep proper record of actual income of each of you, along with expenses. There are certain expenses you can claim from income, for example laptop, internet, mobile phone etc. Consult a CA he will be able to guide and it does not cost much.