I hear you on the tight budget and you have already been given great tips. Personally I prefer fresh vegetables to frozen and I don't eat canned as I find them too mushy for my tastes. I prefer al dente veggies if cooked. I live in a city where veggies can be very expensive especially in the winter season. I value buying local and organic when possible however I buy the best food I can afford. I also do not drive and rely on public transit or walking to do my shopping. The nearest grocery store is across the street and it is not cheap. However even this store has weekly sales and as has been mentioned I check out the flyer every week to plan my shopping. I plan my meals around what is the cheapest and healthiest options.
So this week broccoli is 99 cents a head so I purchased 4. It is not my favourite vegetable but it is cheap. I also got yellow onions, carrots and nappa. Also try the discount vegetable rack. Every two weeks I go to an organic grocery store (I am in the area for an appointment) and purchase from their discount rack only. I often find great deals including sunchokes, yellow plums, rapini and so much more.
There are some vegetables I simply won't be having until the summer like peppers and tomatos. They are just too expensive and where I live they are imported from overseas when I try my best to buy local.
It can be tough but I find ways to prepare the vegetables I get so I don't get bored.