TL;DR
Simply setting ToS or DSCP markings on packets does nothing. You actually need to configure a device to do different things with different markings. If you want that to include queuing, you need to configure queues, and assign different markings to different queues.
A More Complete Explanation
You are wanting to use QoS. QoS is a huge subject, but I will try to explain a few things. A guide to using QoS on Linux can be found at Traffic Control HOWTO.
ToS, which has really been supplanted by DSCP (Differentiated Services Code Point), is simply marking packets to differentiate the various packets for different treatment at some point. The ToS field was part of the original IPv4 packet specification, but there is nothing in the standard that mandates a device must use or respect that field.
QoS involves differentiating (marking) packets, and then doing something based on the marking. What you do with the markings can be things like shaping, policing, queuing (including priority queuing).
A hardware interface will have a FIFO queue, and that is the default and only queue, regardless of packet markings. The hardware is completely unaware of packet headers or markings.
Actually using the markings is usually done in a layer-3 network device, e.g. a router. For instance, you can configure different software queues for a router interface, and you can assign packets with different markings to different queues. Queues are relatively small, not like real buffers. Priority queues will be served before regular queues. Queues don't exist until you define them, and packets are not assigned to different queues unless you have configured rules to do that. You could assign BE (Best Effort, ToS 0) packets to a priority queue, and EF (Expedited Forwarding) packets to a low priority queue.
When a queue fills up, new packets destined for that queue will be dropped (called tail-drop). Tail-drop can be a problem for TCP because it can cause all TCP flows using a queue to become synchronized (global synchronization) where they back off and ramp up synchronously, alternately starving or flooding a queue. There are methods to try to prevent this, e.g. RED (Random Early Detection). RED will actually drop random packets in a queue. This is to force the various TCP flows using the queue to back off and ramp up on different schedules.
Many network switches will automatically assign BE (Best Effort, ToS 0) to anything coming into the switch, unless you have configured the switch to trust the markings on one or more interfaces. Routers will typically trust the markings, but they will not do anything with the markings unless you configure them to do so.
QoS will not work across the Internet, only within your network. You need to have a comprehensive set of QoS policies that are consistently implemented across your network. You may be able to negotiate, for a fee, for your ISP to respect some of your QoS markings and policies, but that only goes as far as your ISP. As your traffic leaves your network, or you ISPs network if you have an agreement with it, your QoS markings and policies will be completely ignored, and the packets will probably be set to BE.
