It depended on the army and the type of musician. Musicians of the period were divided into field music - the drummers and trumpeters that were an organic part of companies and battalions - and regimental bands.
Despite the depictions of 19th century artists, drummers (and trumpeters and fifers) were rarely small boys. They needed to carry a large heavy drum^(1) on the march along with all of their other gear and needed to keep up with a lightly equipped officer on the battlefield. They also needed to keep their heads while under fire and be able to play all the different signals correctly no matter the conditions. This required a great deal of training - one source reckons 5 years to create a good drummer and 10 to create an expert one - however good enough drummers could be trained a few months^(2). Young drummers weren't unknown - the regulation minimum age for Russian drummers was 14, but the Semenovskii Guards reported during the 1807 campaign that most of their drummers were underage; similarly during the dark days of 1814 campaign when the bottom of the barrel was being scraped for the French army, 14 and 15 year olds were hurriedly trained by veterans at the Invalides and sent to the armies - but in normal circumstances experienced men were preferred, the drummers of the British 2/73rd Foot at Waterloo had an average age of 23^(3). Training was usually done the Regimental Drum-Major and his assistants the Battalion Drummer/Drum Corporal, though musicians could be trained centrally - the Russians trained drummers in their Instructional Grenadier Battalions^(4) and France trained cavalry trumpeters at the École des Trompettes at Versailles.
Regimental bands had a more ceremonial role, they would lead regiments on the march or on parade and were seen as greatly beneficial to morale - Napoleon insisted newly formed regiments create their bands before their elite companies. During battle bands would play their regiments into battle and then usually retire behind the main line. There they were used for several different tasks such as collecting the wounded, guarding the baggage^(5) or playing music to steady to main infantry line; as happened at the Battle of Busaco in 1810 when the bands were sent to stabilise the line at a critical point:
Our men, weary and starving were hard pressed by the French, and victory seemed questionable ... [Wellesley] at once sent forward the bands to play the National Anthem, and instantly the men seemed to fight with fresh vigour^(6)
Drum majors would also act as supernumerary NCO's behind the line, forcing shirkers back to line.
Regimental bands by regulation comprised 8 to 10 men^(7), however Colonels would hire civilian contractors as band leaders and to fill out the orchestra - one Russian Jager regiment in 1812 had 2 separate bands totalling 40 men. Colonels could do this out of their own pocket if they were independently wealthy or a collection could be taken up from the regiments officers - one day's pay per month was normal. Colonels would also supply the bands uniforms - the more colourful and gaudier the better for the prestige of the regiment. Napoleon had tried to control the uniforms in 1811 - Soult had noted that the uniforms had gone "from the merely bizarre to the plainly ridiculous" - but without much luck.
With regards to pay, it depended on the army whether musicians were paid more than the standard infantryman^(8):
France - In the Imperial Guard a drummer received 6 francs 90 centimes per 5 days whereas a Guardsmen would receive 5 francs 80 centimes for the same period. A Drummer Corporal, Drum Master Sergeant or Drum Major would be paid the same as a Corporal (8f 30c), Sergeant (11f 10c) or Sergeant-Major (13f 30c) respectively. A bandsmen was paid the same as a Corporal and a band master was paid slightly less than a Sous-Lieutenant (150 francs per month vs. 175). The line was paid less than the Guard, but the overall comparison of pay for Musicians vs. Soldiers was the same.
Britain - Drummers and trumpeters were paid slightly more than privates. An infantry private received 1 shilling per day and a drummer received 1 shilling 1.75 pence per day.
Prussia - Drummers were paid at the same rate as privates (2 Thaler per month), Battalion drummers as Corporals (3 Thaler 12 Groschen) and the Regimental Drummer as a Sergeant (4 Thaler 12 Groschen). Bandsmen (Hautboisten) were paid between a Corporal and a Sergeant (4 Thaler)
Austria and Russia - Drummers ranked as privates, Battalion drummers as Corporal and the Regimental Drummer and Bandsmen as Sergeants-Major.
Notes
1 - Brass drums had become fashionable by the time of the Napoleonic wars, replacing wooden drums due to the richer sound they produced.
2 - The drum-major of the Hexham Local Militia recalled training drummers from scratch in 3 weeks, but this was treated as an exceptional example
3 - The youngest was 19
4 - These were setup to train NCOs, but one company per battalion was set a drummer company
5 - One source states the Austrians used them to guard the regimental colours, which seems risky for the highly valued bandsmen.
6 - "England's Artillerymen" by James A. Browne, as cited in Farmer, Henry George. “OUR BANDS IN THE NAPOLEONIC WARS.” Journal of the Society for Army Historical Research, vol. 40, no. 161, 1962, pp. 33–38. JSTOR, www.jstor.org/stable/44226337.
7 - The bands of Guard regiments were larger due to their greater ceremonial role
8 - 1 Franc = 100 centimes; 1 Pound = 20 Shillings = 240 Pence; 1 Thaler = 24 Groschen
Sources
Swords Around A Throne - John R. Elting
Historical Description of the Clothing and Arms of the Russian Army Vol 10 A - A.V. Viskovatov
The Austrian Imperial-Royal Army 1805 – 1809 - Enrico Acerbi
Geschichte des koeniglich preussischen ersten Infanterie-Regiments - Alexander Carl von der Oelsnitz
Histoire de la Garde Imperiale - Emile Marco de Saint-Hilaire
The Armies of Wellington - Philip J Haythornthwaite