One of the best strategies to inexpensively build your collection is as close as your local bank. There are many coin collectors that start building their collection by searching through pocket change. Taking this a step further (after exhausting all of the change in your pockets) you can head to your local bank to buy wrapped bankrolls.
A few years ago I began buying rolls of half dollar coins, pulling out all the silver coinage, and taking the leftovers to a different bank. When silver was $5 +/- an ounce, I hoarded mountains of half dollar silver coins. Most of them were 40% silver, but I found many 90% silver and even some proof silver coins. Unfortunately most banks have stopped carrying half dollar rolls, but I encourage you to ask!
One of my favorite and ways to search coins is one-cent rolls. I save ANY penny dated before 1982, as the copper inside is worth more than face value. I believe in the future, copper prices will continue to rise and these coins will be harder and harder to find. I have even seen people accepting $50 of pre-1982 pennies in exchange for $100 of merchandise.
Nickel rolls can yield coins that are worth saving. I generally save anything dated 1960 and earlier as they are getting harder and harder to find. Many coin dealers charge a premium for these nickels.
In my opinion, the days of successful searching quarter and dime rolls has for the most part come and gone. This isn't to say that you will never find a gem, but I have found that those discoveries are few and far between.
When people bring in their change, many times banks will roll the coinage straight from counted bags of coins. This leads to rare finds that may have been mixed in with 'grandma's' change when she brought it in. I have also found numerous proof coins and foreign coins mixed in with common bankrolls.
I encourage you to take a chance and see what treasure awaits you!
