You have a telescope with a relatively short focal length. Where this telescope excels is with low-magnification wide-field views.

That's not to say it can't be used to look at the planets, but because it's a fast achromat you'll notice colour fringing on brighter objects. If you find this distracting then you can buy filters to alleviate the issue.

In your original question you asked for "the best" eyepieces, but you didn't specify a budget.

• If money is no problem then don't bother with a Barlow lens. Just go straight for eyepieces at the higher end of the market. Look for the names Baader Morpheus, Nikon NAV, Pentax XW, Takahashi TOE, Tele Vue Ethos / Nagler / Delos.
• If budget is more limited then there are a lot of good eyepieces in the mid-range; names to look for are: APM UFF / XWA, Baader Hyperion, Explore Scientific, Vixen.
• If budget is very limited then get a Barlow lens and then save up for a good eyepiece. I personally use and recommend the Orion Tri-Mag 3x Barlow. The lens element can be unscrewed from the tube it comes in, and then screwed into the bottom of an eyepiece, to give a ~1.5x increase in magnification.

Don't go overboard buying eyepieces - it's better to have a few higher quality eyepieces than many lower quality eyepieces.

As for which focal lengths to buy. When you buy a telescope which comes with bundled eyepieces, those eyepieces have focal lengths which are chosen because they're best suited to that telescope.
The first eyepiece to buy should have the same focal length as the bundled eyepiece you use the most.
A wide-field (70 degrees AFoV or greater) eyepiece will be a great improvement over the bundled ones.

Another thing to take into account is the "eye relief": the distance from the eyepiece lens closest to your eye and the focal point. If you wear glasses while observing then you'll want to look for "long eye relief" eyepieces - over ~15mm is good.

When it comes to planetary viewing there's a problem: planets are very small objects and therefore require high magnifications; but as you increase the magnification above a certain point the image becomes blurry and loses definition.
The point at which this happens depends on the "seeing". A good website which shows the seeing measured in arcseconds is Meteoblue. The lower the figure the better.
At this point you might notice that your telescope's specifications also have values measured in arcseconds. These are the theoretical maximum resolution your telescope's optics are capable of.
What these values basically mean is that planets in your telescope will be quite small and low resolution. You'll be able to see them, but you should temper your expectations.
As for an eyepiece to use for planets: something around 5-6mm will be the sweet spot for your telescope. It doesn't really matter whether you get this focal length by using a 2x Barlow with your 10mm eyepiece or by buying a dedicated eyepiece. Some nights you might be able to push the magnification higher, but those nights will probably be few and far between.

To give you an idea, here's a photo I took through the eyepiece of my 80mm refractor during the conjunction of Jupiter and Saturn in 2020 (pay no attention to the brightness - your eyes will adjust a lot better than the automatic Android camera settings do!): I can't remember exactly what focal length eyepiece I was using that evening, but it was as low as I could go without blurring the image.

What you'll notice, however, is that the rings of Saturn are clearly visible.
You'll find that a smaller planet that's in focus is a lot better than a larger planet that's blurry. With time and practice you'll be able to pick out fine details in a small planet image.