Where would a fresh graduate start with if he/she want to work in the asset management field?

You should approach for the well know Asset Management companies and Brokerage Houses most of them offers MTO position by this you got a good chance to grow and also u come to know well which depart attracts u most due to rotation procedure like investment and settlement department, Account & Finance Department, Marketing and Business development department, camplaince & Shariyah and many more depend on size of company and varieties of Fund. After that u have to do specialization in the field that suits u alot :) 

I'ts great that you started as an intern/learner ship for now.
To start as an intern/learner is a big achivement for a graduate because it gives a graduate a chance to gain expirience and to prove herself/himself. 
  Asset management filed is a very challenging work and it needs a person to take time before he/she takes dicission. 
  First of all form a team of asset called ASSET MANAGEMENT TEAM (AMT) and note that the team is very important when coming to asset management because it will help you to deal with all asset management policies like {ASSET DISPOSAL, PROCUREMENT, DEMAND, MOVEMENT, DISTRIBUTION and MAINTANCE} most of all ASSET MANAGEMENT.
I wish you a good luck in the world of ASSET.
I wish you a blessed of all in your new career.......