You SHOULD be able to see Mars, Jupiter, and Saturn easily with your scope with any eyepiece. In fact, You should be able to see them easily naked-eye. But, I assume you want to see more detail, not just a bright dot in the sky.
Your telescope is a 150 mm Newtonian reflector with a focal length of 750 mm (giving you a focal ratio of f/5). It looks like you have six eyepieces, each with different focal lengths, and a 2x Barlow. Based on that, you should get the following magnifications:
Eyepiece Mag With Barlow
---------- ------- ---------------
32 mm 23x 47x
23 mm 33x 65x
12.5 mm 60x 120x
10 mm 75x 150x
6 mm 125x 250x
4 mm 188x 375x
Realistically speaking, anything over 200 or 250x is going to be a waste in this telescope. The maximum useful magnification of any telescope depends on a fe factors, such as aperture, eyepiece and telescope optics quality, atmospheric steadiness (seeing), humidity, air pollution, etc... While you may often hear people say the maximum useful magnification is about 50 or 60x per inch of aperture, in reality, on an average night, it's more like 25 to 30x. Based on this, you shouldn't expect anything over 200x to be useable very often, maybe 150x, unless you live in a place with unusually clear, steady air (like living at high altitude in the mountains). Actually, I just looked, and Orion says the maximum useful is about 177x, which would be based on 30x per inch of aperture.
Still, with even 177x, you can get some pretty good views of planets. The 4mm will give you 188x, which should be a nice view, though probably a little fuzzy. dropping to 6mm will give you 125x, and that shouldn't be too bad. For Jupiter, you should be able to make out cloud bands and possibly the Great Red Spot, if it's in view. And, of course, you can see the four Galilean moons. With Saturn, you'll be able to see the rings, but I wouldn't expect to see the Cassini division - but you might get a hint that it's there. With Mars you can make out hints of the polar caps, and just maybe a large feature like Syrtis Major. The filters may help with planetary viewing, helping you bring out the details by increasing contrast.
Here's a couple links to useful discussions of filters for planetary viewing:
You will find it easiest to start out with a wider view eyepiece, like your 32mm. Get the planet centered, then switch out eyepieces. Trying to get right-on a target with a high-power eyepiece is much more difficult, even with GoTo.
For other objects, things are a bit different. Deep Sky Objects (DSOs: star clusters, galaxies, and nebulae, basically anything outside of our solar system), are many, varied, and can be quite difficult. Even through larger telescopes, they are often nothing more than fuzzy patches. They will NOT look like you see in pictures.
Your scope has an alt-az GoTo mount. This is not always a good thing. It must be properly aligned, and even then, a lot of alt-az GoTo mounts just aren't all that accurate. When you do your startup routine, it should ask you for at least two alignment stars. You need to get these aligned as exactly as possible. If it gives you the option to add more, do so. The more you add, and the more accurately you align them (meaning centered in the field of view, not just somewhere in the eyepiece), the better the GoTo and tracking performance will be. I've had observing sessions with GoTo scopes where they were off by several degrees. If you don't know your way around the sky to begin with, you have no idea where you're looking.
For DSO's, aperture is important, but so is getting the right exit pupil. A guy I know on Reddit wrote up a great discussion of exit pupil you can read here. Simply put, you should tailor your exit pupil to the need. This partly will involve measuring your own pupil (this Sky and telescope article includes how to measure your own pupil size).
Finding them, then is the issue. Even with GoTo, it's not always easy. And just because your GoTo scope points at the object (if it's properly aligned) doesn't mean you can see it. The description on Orion's Website of your telescope's GoTo capability says: "Huge GoTo database of over 42,000 objects, and handy built-in Tour mode lets you enjoy great views on any clear night." Unfortunately, a lot of these objects simply aren't going to be viewable in your scope, or from where you're observing.
And observing site is an important consideration. If you're in an area with a lot of light pollution, the sky background will be so bright that you can't get enough contrast between it and the object you're trying to view to make the object visible. Still, that said, there are a lot of things you should be able to see from even moderate light pollution. I would start with the Messier objects, as they are some of the brightest DSOs in the night sky and fairly readily found (a couple are even naked-eye objects). There should be a menu option in your scope's hand control for these. If you're properly set up and aligned, it should be able to find them for you.
I mentioned above, if you don't know your way around the sky, it makes it harder. You should definitely take the time to learn the constellations, at least the more common ones, as well as the names of some of the bright stars. Most GoTo alignment systems ask you to locate 2 or 3 bright stars by name, and they mostly use the same stars. There's between 150 and 200 commonly used bright stars for this purpose. I wouldn't say you have to memorize them all, but if you can find a good chunk of them, say maybe 50-75 of the most common, it makes it easier to find your way around the sky. If someone says "it's 5 degrees south of Betelgeuse" or "Find Arcturus and move East a few degrees," you'll have a better understanding of what you're looking for. Also, learning to estimate angular measurements (there's some great tricks out there for estimating degree measurements using fingers on an outstretched hand) can make a difference.
One of the best resources for the beginner is the book Turn Left At Orion by Guy Consolmagno and Dan M. Davis. It's written to help people new to the hobby learn their way around. You can find it on Amazon, Barnes & Noble (bn.com), or order it from any good bookstore (if they don't stock it already).
Probably the BEST resource, however, is to join an astronomy club or society. In the US, membership is usually $50 (USD) or less per year. I suspect they are similarly priced in other countries. They are a great place to meet people with the same interest. The best part is, most such groups have people who enjoy teaching newcomers the tricks to finding things and how to really observe them. They also usually have people who can teach you how to properly collimate, maintain, and even repair your telescope. And a lot of clubs have their own observing sites, or at least have scouted out good observing sites in the area. It's really a worth it to join a club.
My club, the Houston Astronomical Society, only costs $36 per year, and we have some great benefits for members (including loaner telescopes, a dark site for observing, and even small observatory).
Good luck and clear skies!