It’s less about the equipment and more about the photographer. A good photographer can get excellent photos from a phone. A mediocre photographer will get poor results from the most expensive DSLR.
As with many things, the key is experience. If you want to improve your photos, practice. Learn about composition, lighting, movement, etc.
I have a Canon DSLR, my sister has a Nikon DLSR, and my BIL has an Olympus mirrorless. They are all capable of great photos, but we tend to use them primarily if we are on a once in a lifetime trip, if we want to enlarge the photo substantially, or if the conditions might challenge a phone camera. Probably 90% of the time you will see us taking photos with our phones.
Before investing money in new equipment, learn as much as you can about photography and practice with what you already have. You may find that with a little time and effort, you can get the results you are looking for. Phones have a lot of different settings but most people leave them set on the default. Change it up!
There also the category of point and shoot cameras and bridge/zoom cameras. Some of these can take excellent photos, but personally I feel that phone cameras can produce equal results. I do have a Canon bridge camera, but I use it primarily for video where I will be zooming in and out frequently (such as at an air show) since it zooms much more smoothly than my phone or DSLR for that particular purpose. If I didn’t already have it, I would not be buying one these days.
If you decide you want to get a DSLR, any of the big name brands can work well. You don’t have to buy the most expensive model (although you may not want to buy the cheapest model either). It boils down to which one is the best fit for you ergonomically. They will all have the same basic buttons but they will be in different areas, and there can be differences in size and weight. Some may feel better for a left handed person, and others for a right handed person. Perhaps compare a mirrored DSLR to a mirrorless one. You should find a store that carries the brand you are interested in and physically pick them up, even if you ultimately plan to buy online. Also consider how often you plan to use a DSLR. It will be heavier and more cumbersome than a phone. How often do you think you will want to carry the extra weight and accessories like lenses, spare batteries and memory cards? There is a big difference between pulling a phone from your pocket and carrying a DSLR plus accessories.
Have fun, whatever decision you make!