I am putting a bathroom into a basement with no existing rough in.

What is the best way to vent and how far can it run?

The bath is on the wall that runs up to the garage and near the front door. The opposite direction is the back deck.
I would have to run across the joist to get to the side of the house.

You can combine your vents into one, 2" vent. Then run this vent to the outside and terminate it no less than 6" above the roof. In places where you have to run your vent horizontally, maintain 1/4" per foot up-slope.

If you run into existing 2" vent along the way, you could connect to. This way, you wouldn't have to run your new vent all the way to the roof.

Most bathroom groups are roughed in like this.
Toilet connects to sewer main or the stack vent. Lavatory connects to toilet drain and runs a vent off the top the stubout tee out the roof or revents back into a dry vent in the attic.. The toilet wet vents through the lavatory vent and the tub/shower connects to the lavatory drain and is wet vented by it. This is a normal rough in and is acceptable both by local and state codes and also The Standard Plumbing Code Book in 90 percent of the country. Check your local codes to make sure you're not in the excluded 10 percent.. The vent off the lavatory may be run out the roof or revented back into a dry vent in the attic or if you're reventing back into a fixtures dry vent you must make your connection at least 6 inches over that fixtures flood rim. Good luck and thank you for rating my answer. Tom

Thanks both for your responses.
I will be installing a sewage ejector pump and only have access to the main drain before it exits the wall.
Can I use a mechanical Studor maxi-vent or vent out the side of the house?
Both are possible options.

You may use a AAV for the fixture vent but you must run a dedicated vent out the side wall or the roof for the pit. This pit vent must service the pit all by itself. It can not be connected back to the house vent. Good luck, Tom