Pros
No better place to get federal legislative experience, this is where it all happens. For any and all negatives I list, the bottom line is there is no substitute for experience here. Things move slowly, but you have opportunities to contribute to vital legislation that will really change people's lives. If you work in constituent services, you have opportunities to help people individually that will be very rewarding. You can't help them all, but you can do some real good. The staff and members are generally more experienced and mature than in the House. Things move slower, there's more bipartisanship. Pay is publicly disclosed, so little chance for disparate pay/pay discrimination among staffers. Better job security compared to the House - elections are every 6 years as opposed to 2. The Congressional Research Service and other tools are available to help you and your member get to know your issues and your duties. My favorite perk was access to the Library of Congress- basically any book ever written.
Cons
The office culture and opportunities varies A LOT by the boss (the Senator, the chief of staff, the LD). If the boss is good, people will stick around for decades; if not you'll see a lot of turnover. My office was not at all interested in staff professional development or education, I had to make my own effort there. I saw another woman endlessly criticized for needing to leave on time (not even early! on time!) to take care of their kids. The pay is generally lower than what you'd get in the private sector. A lot of bosses will make appropriate pay decisions, but a lot know that an army of 24 year-olds will do the job for free, so you don't have a lot of leverage in pay negotiations. That's why so many people go to lobby after-- it's easy to say they're "greedy" but living in DC is expensive. Office politics rule - the bosses' buddies get the jobs, promotions, raises, period. A lot of opportunities are never even posted and just handed to a favorite. Likewise, an "inner circle" springs up of these buddies, and they make the decisions. If you're not in the group, you're chopped liver. But, they're very defensive and are not interested in changing, so be prepared to play that game or get pushed out. It's a bit like a customer service job - most constituents will be reasonable in their expectations of you and the member, others will not. Some will be very abusive because to them you personify "the government," even if you're trying to help them. Similarly, even as a staffer you have to be very careful about your public/social media conduct as the media/constituents are watching.