You can’t go too wrong with Shimano pedals and shoes. Over 30+ years of riding MTB and trying other brands I always come back to a Shimano / Shimano pairing. I have several shoes (Giro/Fizik/Specialised/5Ten) kicking around with barely 20hrs riding on them as they just never seem to fit perfectly unlike Shimano shoes, although I am sure this is very much down to my wide feet. Shimano has standard and wide toe box shoes (look for the W in the size for the wider toe box) which a lot of European shoes just don’t offer.
Time ATAC is a good system and although rare can be found realitively cheap. However Shimano does have a wider range of budget pedals (speaking of which don’t go for a flat on one side, cleat on the other pedal…) . Another bonus of Shimano S (Shimano) P (Pedal) D (Dynamics) is that you will most likely be able to jump onto a mates bike and clip in as even brands like Ritchey or HT or DMR make 100% compatible pedals now that the SPD patent has expired. You can’t do that with Crank brothers (shitty fast wearing shoe cleats) or Time pedals.
Good luck in your choices. Read and research and I am sure you will make a decision you are happy with.