An altazimuth mount will allow you to point the scope at things, but will not track the objects as the earth rotates - for that you'll need an equatorial mount. Another aspect is that the higher your magnification, the more image jiggle you'll have, unless you have a rock-solid mount.

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It's worth your time and effort to align the finderscope with the center of view of the scope, and to also note whether the image is upside down, mirrored, or something else - then when you move the scope, you'll be able to move it to the object you want to see. The moon is an excellent target to carry out this exercise. With a small scope, it's best to star-hop from a bright object to a dim object. So center the bright object in your viewfinder, and it should also be centered in your telescope's field of view at low power. Use the low power eyepiece to help navigate to the next object, and when you finally arrive at the object you're looking for, you can switch to a higher power eyepiece.