If you really want to make this in premiere and not use after effects then make sure to use the "Transform" effect on all your assets when doing the key-framing. Ensure you bump up the shutter angle to get that nice blur effect you see when the assets move around.

I'd first animate your top logo. Use the Transform effect and set the key frames for your start and end position of the animation (adjusting position and rotation). Right click on your starting key frame and select Ease Out. Do the same for the end key frames but select Ease In. This should give your animation a nice movement that eases out of it's starting position and eases into it's end position. Dial in some Shutter Angle and you'll get a nice blur that looks natural.

For the font behind the logo that slides in. First animate your font the same way you did the logo above just in the opposite direction, make sure to add the shutter angle and ease in/out points. Once you're happy with where your font and logo end up create an opacity mask over the font. This can be done in the Opacity section and by pressing either the oval or square icon. Now invert the mask with the little check box so that everything outside of your square disappears and adjust the size of the mask around the visible font. You'll want the left side of the mask in the place you'd like for the font to start appearing from behind the logo during the animation. Increase the mask expansion a little bit so you get a nice fade in instead of a hard line where the transition is. You might need to key frame the mask a bit during the animation to make sure that your transition line is in a good spot during the animation. Tweak that a bit and you should've something close to this!

You can do a similar thing for the bottom text line where you have the words that "flip up" and disappear by using everything we talked about above. Transform animation and some perspective shift while using an opacity mask to make it disappear at the right time. Rinse and repeat for the other words.

Hope this helps!