I have always wondered what I should do to get items quickly in TF2 without paying any money?
Items are dropped randomly when playing, at the rate of approximately 1 per hour, over a maximum of 10 hours of play per week. There is no way to increase this.
You also receive items from completing the Milestone achievements- the "Earn x class achievements" achievements. There are 3 achievements per class which typically provide you with one weapon for each slot.
Trading is also possible, though if you haven't spent any money you can only receive items- you cannot give them.