Do engines like the GE90 and Trent 1000 use transmissions (planetary gears?) for RPM reduction on the fans, or are they welded directly to the same shaft as the low pressure turbine?
Those with gear reduction are called geared turbofans (GTFs). In general, you can check the Wikipedia article on any engine, see its classification and specifications, and also check the linked citations.
The GE90 and Trent 1000 are not geared turbofans. Wikipedia lists only 5 geared turbofans, with the biggest (a recent engine; first run 2008; entry into service 2016) powering the A320neo family (narrow-body jetliner) and other regional jets.
Because of the immense torque of large turbofans (thrust >200 kN), it is not yet feasible. It took 20 years to develop the ~100 kN PW1000G.$^1$ It took that long to ensure the new components are maintenance free (apart from oil changes). So for the time being GTFs are outliers, and when a turbofan is one, it would be stated.
Note: the fan blades are not welded, rather connected/attached. You can watch a fan removal here (noisy video).
To add to what @ymb1 said, and to expand on your question about "welded", @birdus:
The NTSB accident dockets for Southwest 1380 / NTSB# DCA18MA142, Southwest 3472 / NTSB# DCA16FA217, and American 383 / NTSB# DCA17FA021 all show some great internals about turbofan engines and (mostly) what happens when things fail.
Specifically, Page 15 of the NTSB's Powerplant Group Chairman's Factual Report for SWA1380 has a nice description of the blades and how they attach. It's not 100% clear but reading carefully, it does describe how the blades are attached to the fan disk which is then "bolted to the booster by 24 bolts" (which it further clarifies that, at least on the CFM56, the LPC Low Pressure Compressor is sometimes referred to as the "booster")...
That said, the CFM56 and GE90 do share a common characteristic: the LPC (Low Pressure Compressor - aka "N1") actually has compressor stages attached to the same shaft as the fan.
The Trent 1000 is pretty different - the entire Trent series really evolved from the venerable Rolls Royce RB211 and ever since then the entire series has been a triple-spool turbofan engine (as opposed to the twin-spools utilized by every(?) other manufacturer.)
So on a RB211/Trent XXXX, you have 3 independent spools and 3 gauges in the cockpit: N1, N2, and N3 - corresponding to the LP, IP, and HP shafts (low/intermediate/high). The whole idea of this being to decouple all of the core compressor stages from the fan rotor. If you look at the various compressor stage/turbine stage configurations, the notable difference with a Rolls Royce RB211/Trent is that the Fan has its own LP turbine.
The Trent 1000 it has "Compressor: one-stage LP (fan), eight-stage IP, six-stage HP compressor" and "Single-stage HP turbine (13391 RPM), single-stage IP turbine (8937 RPM), six-stage LP turbine (2683 RPM)". So that means that there's a 6-stage turbine at the back that then spins the one-stage big fan at the front.
The entire purpose of the three spools - even though it is far more complicated than having 2 - is to allow the fan at the front to rotate at a different speed than the engine core compressor sections. Doing so allows you to run the core faster/hotter without spinning the fan at the front too fast.
The PW1100G from Pratt & Whitney is their answer to the problem of needing to turn the fan at the front at different speeds. Instead of using 3 spools to accomplish the task, they use a gearbox. This allows for different fan blade geometry and allows for a much quieter engine. (Speaking from personal experience, the PW1100G on the A220/CS series is seriously quiet.)
Not to be left behind, Rolls Royce is developing the "UltraFan" which will apparently vary the pitch of the fan blades during flight.