Disclaimer: I am not a professional plumber. I am a DIYer who has been studying plumbing code and learning about how things work. I have learned quite a bit from the pros here, but I know that I still have more to learn.
Hmm.. Well, I see some problems with the floor joists that are notched at the edge. That just ruins the structural integrity. If I remember correctly, you are not supposed to notch a joist on the edge- it should be notched in the middle and not be more than 50% of the thickness of the joist. I think you are not supposed to notch within 2" of the edge of the joist either (but I could be wrong on those numbers).
https://www.familyhandyman.com/floor/how-joists-work/view-all
Are you planning to use some perpendicular boards to box in the notched joists for extra support?
As for the piping, that is quite a mess there. Are they sloping downward at 1/4" per foot?
In order for the vents to work properly, you can't go vertical on the trap arm (the part after the P-trap) until you have the vent. The developed length of the trap arm must be no less than 2 pipe diameters long (so if you have a 2" pipe the arm needs to be at least 4") and no more than a certain length (which differs according to the plumbing code you use-- IPC lets you have about 6 feet for a 2" pipe, UPC is shorter (my state uses IPC so I'm not sure on UPC rules). The drop on the developed length of the trap arm can not exceed one pipe diameter (so a 2" needs to have less than 2" drop). The purpose of minimum length is to prevent a crown vent that would allow water to be siphoned from the trap (and make it so it would not block sewer gases from coming up). The maximum length is to prevent the arm from filling all the way to the top- thus blocking air and creating a vacuum which would make it harder to drain.
Once the vent is reached, the vent pipe must slope up at 1/4" per foot so that any condensation will run down.
How far is the nearest wall that you could run pipes in to? Where is that vent pipe going and how long is it? Is there a way for you to move the drain to that wall instead of under the tub?
If so, keep the tub drain trap in that direction but take off the bends and make it go straight to the wall and put the vent up in that wall. Take the trap on the shower and flip it the opposite direction and run it's vent straight to the wall. If you are running the new vents to meet with an existing one, you would run both tub and shower vents vertically until they were at least 6" above the flood level of the fixtures and then you could run them horizontally (with the upward slope) to meet the existing vent pipe.
From what I'm seeing in your layout you have the tub drain going vertical before it reaches the vent. Those are some huge notches in the joist which will require more reinforcement (basically you may need to box the joists).
Looks like you used a tee-wye for the sweep to the horizontal vent from the tub. The angle is such that I don't think it would block the air from the shower. Generally wyes are not used in vents because the angles are such that they block the air- but I don't know how it would work in horizontal application. Additionally, you have the vent line going horizontal under the floor instead of 6" above the flood level of the highest fixture-- this means that the vent can fill with water and block the air. So, basically neither fixture would be vented properly.
What fitting did you use to make the tub's drain go vertical? I can't see it very well and it looks like it is some sort of double elbow type thing.
A sketch and dimensions of the room as well as more pictures of the surrounding area would be helpful.
To the pros: If I'm wrong about anything, please let me know.