Simple (autonomous) GPS is free, and depending on the reciever will be accurate to about 10 meters. You can purchase various types of corrections to improve on this ranging from WAAS, DGPS to NRTK depending on the capability of your receiver and how much you want to pay. Sub-meter broadcast corrections via WAAS are available on many mid-range receievers, whereas you will require a survey grade reciever and RTK or NRTK corrections for centimeter accuracy. The accuracy of an acceleration signal is based on positional accuracy and how often your position is updated, see NMEA 0183 for data formats. Some corrections are better suited to moving observers, e.g. RTCA is designed for aircraft whereas RTCM is designed for marine use. On survey grade receivers you can typically configure the dynamics of the unit to be static, slowly moving or quickly moving to suit your application.
Edit: With regards to other comments about the Americans turning the signal off, this is highly unlikely, as the russians also have actively maintained sattelites out there (Glonass), and europe is putting more into space (Gallileo). Newer receievers, often dubbed GNSS and G3, combine corrections from all these constellations to improve accuracy and reliability.
Edit 2 Using autonomous GPS without any other sensors, time tends to be very accurate and position much less so, so a good way of getting velocity is to model the probable dynamics of your movement, and use that model for smoothing purposes. e.g. If you join the dots on raw GPS position you get a zig zag line, whereas if you are driving in a car you know that you are generally travelling in a reasonably straight line, and changes in acceleration are similarly smooth. Thus if you take your recent positions, and do a least squares line fit, arc fit, or similar curve fit through them, you get the most probable line travelled. You can use this line to get a better running average determination of velocity. If you have background mapping of a road, you can also use that for regression analysis and velocity prediction. If the average direction you are travelling is varying wildly in this scenario, and the position isn't changing much, chances are that your are static. In a ship or plane the dynamics are different, you might have much larger radii that you can possibly turn through, and can factor these into your analysis. If you are walking, you can zig zag all over the place, although you usually don't. For this reason, many hillwalking GPS units include a magnetic compass and correct for bearing.