Jon -
As one who lives in NYC and uses solely public transportation to go to Rutherford and Kearny and other places in Jersey I might be able to help.
No Driving -
1)
Multiple stops along the way for the bus you are referring to the 13. The good thing about that line is some buses at rush hour run limited stops when going through Newark. Probably best choice if you want to avoid driving. Also this route takes you by Military Park which is fantastic since they fixed it up and the redeveloped areas of Broad as well as some of the cultural attractions in Newark - NJPAC, Prudential Center, Newark Museum, Ironbound. So great option if you want to do something without going into NYC. Here is a bus map for Essex Country which Belleville and Newark belong to -
http://www.njtransit.com/pdf/bus/Essex_County_Map.pdf.
Also useful as the towns across the Passaic are either in Hudson or Bergen are the bus maps -
http://www.njtransit.com/pdf/bus/Bergen_County_Map.pdf.
1A) You could take the 13 the OPPOSITE direction to Clifton Commons and catch the 192-193 NJTransit bus to NY Port Authority Bus terminal in Times Square
2) If you walk south to Belleville Ave and at the intersection of Union/Belleville Pike catch the DeCamp Bus Lines 44 to Timnes Square NYPABT (it is a private bus line and the fares are higher about $6.50 one way to Port Authority). Bus Routes & Schedules from Northern NJ to NYC. It runs hourly on the weekends and every 30 - 40 minutes or less during rush hour. No overnight service. Last run from NYPABT is about 11 pm.
Unfortunately no jitney service to Washington.
DRIVING
1)
Yes you can drive to the light rail at Grove Street or drive to Newark Penn if you want to save time. As I see you aren't too far from Belleville Ave, you could, when the weather is good drive to Washington then turn onto Washington and turn right onto Rutgers for the Bridge to Kearny and then Jersey City. The bridge takes you onto the Belleville Pike which then takes you over the bridge to Jersey City. Bear right to get onto Kennedy Blvd and turn right to Journal Square. Not sure I would do this all the time but definitely worth trying out as the nice thing about Journal Square is you can get to 34th/Hoboken/WTC/Newark without transferring to another train. Plus there is a shopping center at Belleville and Schuyler at the Kearny/North Arlington border and excellent rice and bean bbq place called the Firepit near there and there are a few good restaurants in the Journal Square area. Probably a good option if you are hanging with friends in Jersey City after work.
Another option closer to Newark Penn is Harrison PATH which you can catch off McCarter but take the bridge to Harrison at Harrison Ave and then head south on Frank E Rodgers BLvd. nearby is Red Bull Soccer stadium so if you are into soccer and sporting events good place to go to. North of this area off the Clay Street bridge in Newark heading into Kearny is a BJs - a sort of older style Costco opening up August 27th and LA Fitness with a nice pool. You have Planet Fitness near you and a few other gyms, so this just adds to your options.
The main issue with driving to Newark Penn is the traffic can be terrible just where 280 crosses Broad above Newark Broad Street Station due to ongoing construction. So the light rail would be best unless the light rail is experiencing delays. Then review the options I gave.
2) A bus option a very short drive from you and a good resource for you if you want to have a movie night out, access to affordable dependable chain restaurant and Barnes and Noble/Starbucks after work - Clifton Commons just north and west of you at the Bloomfield Clifton border. Its a mall served by NJ Transit 192 and 199.
3) If you drive to Bloomfield you could catch the GO28 bus to Penn Station also, a limited stop bus serving Bloomfield, Newark and Newark airport. It is a NJ Transit bus.
4) Trains to 34th Street - aside from Newark Penn in Newark, Secaucus Junction, Secaucus, and Bay Street in downtown Montclair that offer quick or express service to NYC - there is Watsessing Ave at the Bloomfield/East Orange border and closer to you the Delawanna station in Clifton I believe - most run to NYC but some terminate at Hoboken, Secaucus, or Newark Broad.
LATE NIGHT SERVICE
1) Options are limited, but there is the 190 bus stopping in Clifton and Rutherford and Lyndhurst Office Park. It runs 24 hours with only a 1.5 hour gap between 4 - 5:30 am. From there you could grab a cab and save a little money compared to taking one from Newark.
2)The PATH is your cheapest PUBLIC TRANSIT option BUT you have to grab a cab from Newark, Harrison or Journal Square. Newark would be the cheapest as you would just go up McCarter to hook up to Washington. If you want to feel completely safe, go to Harrison and when you walk down the stairs there will be a few cab companies either right by the entrance or a little south. The cab can then drive over the Jackson street bridge to hook up to Market in Newark and then connect to McCarter to Belleville. (Or they might take Passaic Ave in Harrison/Kearny, cross the bridge to Belleville and run along Route 21 to catch Joralemon if they don't feel like driving through Newark.