This is a little strange. Usually a new user will just create an account based on his phone number (including the country code. Eg +1 for USA and Canada). A verification code will be sent to that phone number and you will need to input that to proceed the registration. Then, you choose a password for subsequent logins. AFAIK, the new account username will (initially) be the phone number supplied by the new user.
Anyway, even with a random username generated by Taobao, you can still use it to login and then change it to whatever you want.
Have you already passed through the step of receiving the verification code and then supplying a password for your account?
If you forget your password or have not set it up initially, you can click 'forget password' while logging in and the system will send you a temporary one to your registered phone number for login purpose.
Regardless, there are 2 alternatives to using Taobao directly.
1. You can use Aliexpress
Type LZmode A24 V5 or ZS-D2 and you will hit a lot of responses from different shops.
Pros:- most of them can communicate fairly well in English, relatively to Taobao
Cons:- prices are higher. Taobao is really setup for the local Chinese market and so prices are the bare lowest. Aliexpress is setup mostly for the international market and so you get 'regular' prices.
2. You can use an buying agent (eg Superbuy)
Cons:- you need to pay an additional agent fee
Pros:- they pretty much handle the whole process for you. I have never used this method as I mostly buy directly from Taobao.
But many members here have and seems to say Superbuy is a good agent. Ask around.