Make a sextant from a protractor and a piece of string about a foot long. Tie one end of the string to the hole in the center of the protractor. When the bottom of the protractor is on the ground and the string points to the top of an object, hold the string against the protractor and read off the angle of elevation. Hold the protractor sideways to read off the angle to the ends of objects like bridges.
One easy trig problem is to compute the height of the flagpole. Pace off one hundred feet from the base of the school's flagpole. Measure the angle to the top of the flagpole. If the angle is alpha, the height of the flagpole is equal to 100 times the tangent of alpha. This is because the tangent of an angle is equal to the opposite side divided by the adjacent side.
If your schoolyard does not have a stream, it probably has some other feature that would be hard to measure such as a gully or ravine. Find a prominent feature -- like a trashcan -- on one side, then pace upstream 50 feet on the other side of the feature and measure the angle across the stream to the prominent feature. The width of the stream is 50 times the tangent of the angle.
Sometimes it is hard to find one side of the triangle. For example, if you wanted to find the height of the school building but you could not pace off a distance because there were bushes all around the building. You can still find the height of the building if you pace off a distance along a line perpendicular to the building and measure the angle to the top of the building at each end of the line.
You can assign the problem of finding the height of a building if you know the two angles of elevation to the roof from different ends of a perpendicular line as a difficult homework or group in-class problem, or you can just give the students the formula. It is h = d/(Cot A - Cot B), where h is the height of the building, d is the length of the line, A is the small angle and B is the large angle.