I'm not sure if Mountain Bikes work the legs out in the same way. Mountain Bikes give you more room to move around in comparison to this type of bike
Mountain bike geometry lets you move the bike around under you more, the extra clarance afforded by the lower top tube lets you lean and sling the bike around.
A road bike with dropped bars gives you far more body position options. You can ride on the flat, on the hoods, mid drop, full drop, and places in between.
So an mtb is more critical in a way with setup of bar position (height and length) to avoid lower back pain.
some padded shorts
I detest padded shorts, and padded saddles.
I'll only use leather saddles (brooks mainly, that type) and seamless shorts/bottoms.
Any padding increases pressure on the perineum (your sit bones sink into the padding allowing the middle to rest on it, bad). If you must have padding, get things with a hole in the middle.
cycling shoes
You can push into the pedals in multiple directions. Except for backwards
Get SPD (mtb bias) or SPD SL (road bias) (shimano names) shoes and the pedals to go with them.
Then, you can do the traditional 'push' pedalling but you can also pull backwards and upwards without losing your footing.
IMO, clipless pedals (which is a stupid name, because the entire point is you clip into them, but I digress) are the best thing you can do. Not only do they hugely increase efficiency, they allow you to use muscle groups that are usually ignored with cycling.