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What's in a name?
The first thing I would do would be to bring back the Enterprise, except I would also take some of the glory away from it; perhaps the previous Enterprise crew  was responsible for some horrible treasonous acts. The previous Enterprise crew suffered from the weight of history; constant comparisons to the crews that came before broke their spirits and pushed them into treachery.

As a result, the Enterprise name was retired from active service, and the new crew is taking the mantle back into the cosmos because of a special request from a powerful politician. 
Contract Bridge
I would also bring back the binary helm/navigator. The astrometrics position (from Voyager) developed a new necessity for a position dedicated to spatial phenomena, and the navigator fulfills it nicely. The helm still serves the same function, and needs no change.
There's always going to be an ops position, but that always seemed to be a made-up position that nobody would really need to do.  Isn't that why they have computers?   Tactical is important, since there's always ship-to-ship combat.  I think a transporter pad very near the bridge is important, just like they had on DS9.  I'm envisioning a loft bridge, with the loft housing the ops/science/engineering stations we saw opposite the view screen in The Next Generation, and the transporter pads.  I think a really big viewscreen would be cool, if only because nothing says the future like big TVs.

 

 

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  • Who Needs Pads?

After having given it some thought, it really makes no logical sense to have a transporter room.  150 years after the current timeframe, it is completely possible for the transporter technology to have advanced to the point where site-to-site transports are the standard.  In the Original Series, the crew had to press a button on a console device in order to communicate with other parts of the ship; in the Next Generation, technology had developed into a decentralized system accessible by voice from any location. 

In my series, the transporter technology is similarly decentralized.  Anything can be transported anywhere else from any location within range of the transporters. 

The original reason for a transporter room was so that characters could have dialogue scenes between the bridge and the room.  This is no longer necessary.