Hooray! Not a lot of love for English Civil War on this subreddit. In general, I guess the best book to read is Hutton's 'The Royalist War Effort', which particularly focuses on the early war, as well as Morrill's 'Revolt in the Provinces'. I'm also happy to provide more sources (or just PM my dissertation's bibliography) if anyone is interested.
Anyway, the answer is complicated and varied. In general, however, the organisation proved inferior to that of the Parliamentarians, both in the Royalist 'main force' based around Oxford, and in the overall Royalist war effort that spanned the entire British Isles.
Initially, the war was quite focused: Charles raised his standard at Nottingham with his supporters, and it was broadly assumed that a single big battle (which eventually took place at Edgehill) would settle things in the first few months. Things were very haphazard: the efforts of his closest friends, family and supporters would be crucial in creating the military force that would fight at that battle. For example, his nephew, Rupert of the Rhine, was largely responsible for organising the cavalry force, and Jacob Astley worked hard to create an infantry core. Charles himself proved willing to work hard at using his royal persona (and that of his family) to raise support.
The soldiers in the army came from many sources: some were picked up from 'Trained Bands' (militia organisations of variable quality) along the way, others were recruited by commissions to officers to raise units at individual villages (these individual commissions were much more successful than the use 'Commissions of Array', a formal method of raising support dating back to the Middle Ages that the Royalists utilised). Others volunteered, were hired from overseas, joined with their noble masters, and came from a variety of other sources. They would generally serve in a regiment or squadron organised by their recruiter (regiments typically bore the name of their commander/recruiter, e.g. 'Lisle's Foot').
After the inconclusive (though bloody) clash at Edgehill, and the stalemate that followed, things spread out a bit, as it became clear that the war could be prolonged. Royalist and Parliamentarians had started to raise support and skirmish from the start of the war, but it now became clear that the whole of England and Wales was to be a warzone. Controlling such as large war effort across a huge area was far beyond the ability of Royalist central command (based in Oxford). In areas such as the West and North, local commanders essentially fought their own war, independent of central organisation, with interaction rare (a theory by Gardiner that there was a broad strategy of a three-pronged advance on London has been thoroughly debunked). For example, in Yorkshire the Earl of Newcastle was almost entirely at his own discretion for how to advance the Royalist cause. As such, it is hard to talk about a general 'Royalist organisation' as this varied from area to area.
This decentralised structure was heightened by the nature of king Charles's command: in war, as in peace, he proved to lack even the most basic skills of leadership and management. He proved overly deferential to subordinates, and attempts to exert his authority usually proved disastrous. It can be broadly shown that he took little role in the organisation of his forces, nor in the strategies that they embarked upon. Charismatic figures, such as Prince Rupert, were able to do as they pleased, and central command usually proved to be a paranoid, bickering and ineffective mess.
Administrative systems had to be organised individually for every region, due to the impossibility of a central logistical centre due to distance and scale. The Royalist finances were equally haphazard and decentralised (Engberg's journal piece is a great place to start on this topic). Commanders were typically starved of pay, and had to make do with their own provisions, often supplemented by inevitable looting and other such abuses (wars were ruinously expensive in this period, and so struggles were inevitable in any case). The same could be said for supplies: regarding money and support, the system was firmly 'pull' rather than 'push', with the most vocal and politically-fluent figures finding sly ways of supporting themselves at the expense of their colleagues. Broadly, administration was strained and inefficient.
A big characteristic of the Royalist organisational style was 'legalism', which Morrill has written a lot about. Broadly speaking, it was thought that every effort should be made to avoid breaking the laws of the land, for perceived moral and practical advantages. In the end, this proved disastrous: unlike the Parliamentarians, who proved willing to bend the rules where necessary, this was very inflexible, leading to obstructive policies. Furthermore, the commanders that did feel the need to break laws had to do so at their own discretion, leading to more indiscipline than if a flexible approach had been adopted in the first place!
TLDR military units were organised from a variety of sources, largely under the control of individual recruiters. Overall organisation was highly decentralised, varying from area to area to the large degree of independence of local commanders. It was often extremely ineffective, due to, amongst other causes: the large scale of the war; Charles's weak leadership; poor administration; and legalistic policies.
(Obviously this is a very broad-brush approach, and there are a lot of things I've left out, so if anyone has any further queries or wants me to expand/give more evidence, I'll do my best, probably tomorrow.)