The best way to design a free construction logo would be to use a graphics editing program like Adobe Photoshop or Adobe Illustrator. If access to those applications is not available, there are several free alternatives.
For beginners, Photoshop and Illustrator may prove difficult to use. However, there are a huge variety of tutorials available online, many at no cost.
Of course, these applications only allow you to design the logo. They won’t make it themselves. The details -- what it actually looks like -- are up to you.
Consider exactly what the logo should look like – what values it should convey. Look at logos used by similar companies and pick several favorites. Then, try to figure out exactly what it is about the logo that appeals. Is it a certain picture? The font? The color scheme? All of those elements can be incorporated into the new logo.
Start simple. Pick out a font for the company name and work from there. Try out a few designs and see which one fits best. Ask for outside opinions. There’s no sure-fire way to make a great design, but with a little work, most people with basic technology skills can design a logo.