Is it possible to see Mars, Jupiter and Saturn through the National Geographic 76/350 Dobsonian Telescope? It's the most affordable telescope I can get at the moment. Also, what about the moon? Will it look small?
Specifications
I see that you have correct data for the magnification. Possible magnifications: 18x, 36x, 58x, 117x. The rule of a thumb is, that for optimal observing of the planets you set the magnification of 20x to 30x for every inch of the aperture. Your aperture in inches: 3. That means, that the optimal magnification is from 60x to 90x. But this rule applies to telescopes with better optics. I don't know how good the mirrors are, but I would recommend the setup with 6 mm eyepiece without a Barlow (= 58x). With this setup, you can observe the disc of the Jupiter, but also the rings of the Saturn. You will be able to see main craters and mare on the Moon.
If you place the Barlow on the scope, the magnification is too high for the small aperture. You should consider it only when observing near double stars. Magnification of 117x won't be useful with planets. Why? Look at the icon in the title bar of this Stack Exchange. When it is small like it is now, you can resolve the big letter A. But if you expand it to the whole screen, you would see only pixels and some form of A. Smaller picture is better, but not too small. Let this rule guide you.
So what is the answer? You will be able to see the Moon and its mare and some craters, but also Saturn with his rings. Jupiter's disc will be seen with ease, but harder will be Mars just on peaceful nights. What about Venus? Maybe better than Mars, but just small Moon like form. Mercury will be visible only as small dot. Neptun and Uranus are too dim to be observable with this telescope, but maybe you will be able to see them with some experience after 1 or 2 months (only like normal stars, maybe blueish). If Pluton is planet for you, it is far beyond the range of your scope. Even I couldn't see it never with my aperture of 250 mm.
But planets aren't everything. If you have budget for buying the solar filter (cheap ones for 10 dollars), you could point the telescope towards the sun, but beware! Don't mimic Galileo Galilei, who was observing the sunspots throught the telescope when the sun was low without solar filter! Check the page https://spaceweather.com/ for sunspots. Also, you could look at the globular clusters with the smallest magnification. Try Andromeda galaxy! It won't have some visible form, but it will be some cloud of brighter region. You could try also nebulae, for example M 57 in Lyra.
You see that you can do a lot with this telescope, but it won't be some spectacular sight and these are the only things you could observe. For more, you have to buy the better telescope. Hope you will have fun with it!
Yes, you can, its just a matter of finding the right magnification, data, trajectory, and where to look of course, But it is possible.
