Trade agreements consist of agreement on tariff levels on the goods/services involved (which should be easy), the standards which the goods/services need to meet (more complex, where these differ between the two parties), and the regulation/customs processes surrounding buying and selling stuff. With the EU at present we have zero tariffs, harmonised standards and a customs union. The EU agreements also govern the quality of what is imported from outside the EU and the tariffs (only 3% for the US goods - virtually no food because of GM - at present). Presumably we will want something similar with all of our future trading parties. On the Standards stuff, which is very important for us consumers, it will be very complicated. Or we could just adopt EU standards as ours, which would make things easier, but still exclude certain products/services from certain countries.
One thing which (probably to your surprise) I think we should do is politely tell the EU to **** off with its 'you are not allowed to formally negotiate bilateral trade agreements while you are still in the EU' line. This is an unprecedented situation, no country has left the EU in this way before, as soon as we invoke Article 50 we will be going and we have the right to look to the future, especially as we know most of these agreements take a minimum of three years to sign off. We would not implement any agreements until we have left, but should be free to set them up (though I doubt we have the capacity and competence to do many).
The thing which confuses me is that free trade is what the big business globalists like, and I thought many Brexit voters don't like them. Surely a more protectionist stance would be better for the core vote, at least in their opinion. Is the UK government both pro free trade but against the impact of globalisation on British citizens?