I live not far from 290 near Cypress, and I'm assuming you're coming from the Brenham area. If going south to I-10 via a detour on 36 isn't feasible, I'll offer some thoughts.
As mentioned, 290 is and has been under construction for quite some time now, and even we locals can get surprised from day to day by the changes. And almost any Saturday/Sunday driving requires advance research because, with some frequency, periodic closures--sometimes of major interchanges or even an entire side of the highway--providing unwanted surprises.
But 290 doesn't intersect with I-10, so you won't be exiting that way no matter which route you choose.
On the plus side, if you want to come in via 290, opening at 10:00 a.m. right at the Jones Road exit is the Freedom Munitions retail store, and they're having a 5% off sale on all in-stock pistol ammo.
As others have mentioned, the Grand Parkway (99) is toll-tag only, no toll booths. The Sam Houston Tollway (the Beltway), on the other hand, is not mostly toll-booth free, I'm hardly ever on the east side of town on the Beltway, but I travel it fairly regularly from I-45 in the southeast, all the way on the southside, the westside, and the stretch north from 290 to I-45 and can attest that there are manual-pay toll booths as well as no-stop toll-tag lanes all the way along. However, if you're going any distance, the sheer number of toll booths will be off putting. When I drive the PSC range, I go through six toll stations each way...without an EZ-Tag, I'd never go that route.
Another comment about I-610. It basically is the city center anymore, with the possible exception of its east and northeast bands. Anything I can do to avoid 610 West weekdays between 7:00 to 9:30, 11:00 to 1:00, and 3:30 to 7:00 (2:00 to 7:00 on Fridays), I do it.
I'd say you have two choices.
Option 1: take 290 to Beltway 8, then the Beltway south to I-10. You'll brave a longer stretch of 290 and its sometimes odd-looking lane structures to Beltway 8. The exit to the Beltway will be on your right after Jones Road. A new lane feeds into the Beltway at West Road, but that one is a forced exit at Jones. Another new lane is created at Jones, and that's the one you want to merge into. You'll curl right onto the Beltway.
The pretty consistent rule of thumb is that the three left-hand lanes will be toll-tag only at the stations. The manual-pay lanes are to the right. Some of those may be exact change only, so if not equipped or need a receipt, be sure steer farther right into an open attendant-occupied station. Merging back into the flow of Beltway traffic after manually paying toll must be...interesting when the traffic is heavy (when is it not?) and moving at a clip; I dunno; I haven't paid manually in over a decade. And FYI, the Beltway is posted at 65mph, but if you adhere to that, expect traffic to pass you like you were doing 55. That's one thing that makes merging so much fun.
The good news is that there's only one toll station between 290 and I-10. The exit to I-10 will be six miles from the time you start your curve from 290 to the Beltway. The exit will be on your right. The far right lane will go to I-10 westbound; the one next to that will curve left and feed into I-10 eastbound. That merge can be a bit of fun, too. You'll meet traffic coming in from the Beltway northbound, then both of you will jockey for position in the single lane--the one you're already on--that feeds onto I-10.
Entrance onto the freeway propper will be easy because a new lane is created there. But start moving to your left right away: that new lane is a forced exit at the next exit, Gessner. Depending on how you count, I-10 is up to nine lanes wide in this stretch. Five miles down the road, you'll come to the 610 interchange. Sounds like you intend to keep going east, so pay close attention to the lane markings you'll see shortly after the Chimney Rock exit. The four right-hand lanes are forced exits to 610, the far two lanes got 610 south, the next two to 610 north. The four interior lanes (not counting the toll-tag only lane to the far left) continue on east on I-10. I would recommend you not be in lane five, but in lanes six or seven. There is inevitable conflict in the innermost I-10 lane with the 610 lanes because people are either on their cell phones and forget to move over for 610, or they're trying at the last minute to make a mad dash into the I-10 lane because they forgot/didn't realize they were four lanes over and
still in a forced exit lane.
Option Two: exit 290 at Highway 6 southbound; travel 9 1/4 miles south to I-10 and climb on eastbound. Slower with umpteen traffic lights the first five miles. The final four miles is through the Addicks ACOE land and you'll encounter only only a couple of lights until you come up on I-10.
There's even a bit of a shortcut to avoid the more hectic 290/Hwy 6 interchange:
exit at Huffmeister, hang a right, and follow Huffmeister through a more residential area for about a mile until it intersects with Highway 6; hang a right onto Hwy 6...you'll have saved yourself a little traffic congestion plus about 0.8 miles of concrete.
Near the tail-end of that leg, there'll be an intersection with Patterson Road to your left, then unbroken ACOE land for about 1.4 miles before you come to the exit to I-10. Counterintuitively, you'll be exiting to your right, not the left. The left lanes continue over I-10. After that right-hand exit, stay to your left. After you cross under I-10, you'll turn left onto the frontage road and stay left to feed onto I-10. That entrance forms a new lane which is a forced exit, so you'll want to move to your left another lane after entering the freeway.
Ta da! A whole lotta words to describe two relatively simple routes. But I've lived in Houston continuously for the past 25 years. Trust me: even locals don't like to drive unknown routes.