Right off the bat I'm going to say my answer comes from James McPherson's Battle Cry of Freedom, because it is a great one-volume history of the Civil War era, and because I was just re-reading the part about conscription a few days ago.

As far as conscription in the Union went - the draft was first authorized by Congress on March 3, 1863 under the Enrollment Act. It authorized the War Department to set up a Provost Marshals Bureau, which would dispatch marshals to each Congressional district in order to enroll every male citizen and immigrant who had filed for citizenship between the age of 20 and 45. This enrollment list was the basis for each district's quota in call-ups authorized by the President (four call-ups would occur, one in July 1863 and three in 1864). As a side-note, the reason conscription was authorized (after long debates and heavy Democratic opposition in the 1862 mid-term elections) was because the all-volunteer army was hitting a number if serious expiration deadlines in 1863: 38 two-year regiments raised in 1861 were having their terms of service expire, as were 92 nine-month militia regiments raised in 1862.

Anyhow: in the 1863 draft, provost marshalls called up 20 percent of enrollees, chosen by lottery. In the 1864 call ups, the districts were assigned a pro rata share of total draftees requested by the President, with adjustments made for those had already enlisted from the district. A district had 50 days to meet the required share with volunteers, and if that target wasn't reached, a lottery draft was held to meet the quota.

If you were a man of age and your name was drawn in a conscription lottery, McPherson notes that the least likely outcome was that you'd join the army. Of 776,000 men called up by lottery in the four drafts, a fifth (161,000) "failed to report". They just didn't show up. This could involve skipping town, hiding in the woods, or heading West or to Canada. For those who did report to the provost marshals, an eighth were immediately sent home because district quotas had been filled.

Of the remaining 522,000 called up for service, a whopping three-fifths "were exempted for physical or mental disability, or by convincing the marshal that they were the sole means of supporting a widow, an orphan sibling, a motherless child, or an indigent parent. Unlike the Confederate Congress, Union lawmakers allowed no occupational exemptions."

So basically the most common way of getting out of the Union draft was to claim family circumstances, that you had dependents relying on you for their living. This was not only the most common way of getting out of the draft, but basically the most common experience period for anyone called up.

If you really couldn't convince a Provost Marshal about your fitness or family situation, there were still two last methods to avoid being inducted: you could pay a commutation fee of $300 (which was not an insubstantial amount of money at the time and only commuted your service for the current draft you were called up for, not necessarily future ones), or you could hire a substitute (which exempted you from this and any future draft). Interestingly, the commutation fee was introduced as a means of putting a cap on the potential price of hiring substitutes and keeping it within the reach of lower-income men, while maintaining what was considered an acceptable tradition of substitutes. Commutation was repealed in July 1864. Up to that time, often the fees were covered by local governments or political machines (like New York's Tammany Hall) paying the fees out of property taxes or donations, or from joining draft insurance societies.

The actual implementation of the above varied wildly depending on the provost marshals involved. Many itinerants were simply missed from enrollment. Some districts had padded enrollment lists with fictitious names (the marshals involved basically decided to just draw a paycheck without doing the actual work required). Other marshals were pretty bluntly intimidated into not gathering names in areas that tended to be pro-Democratic and hostile to the war and its aims - these included many of the southern parts of Illinois, Indiana and Ohio, coal-mining areas of Pennsylvania, and parts of New York City. Governors and Congressmen would also fight over enrollment lists, and on more than one occasion district enrollment needed to be scrapped and redone.

Even with enrollment completed, for those called up, there were all sorts of holes in the system with room for fraud, like bribing a surgeon to claim a physical disability (and remember this is before there was a licensed medical profession), filing false affidavits about dependents, faking a physical or mental disability, claiming to not be a citizen, or self-mutilation. And this is all assuming you couldn't straight-up bribe the marshal to be excused.

Of the 207,000 men who reported and were not excused, 87,000 paid the commutation fee and 74,000 hired substitutes (which mostly were 18 or 19 year olds or immigrants who had not filed for citizenship, ie military age men who were not in the draft enrollment lists). Only 46,000 out of 776,000 actually personally joined the Army after being drafted.

As McPherson notes, this wildly inept, clumsy and sometimes fraudulent conscription was ultimately a means to stimulate volunteering and re-enlistment. Compared to 46,000 drafted and 74,000 hired substitutes, some 800,000 volunteered or re-enlisted during the two years the Union draft was in effect. Much of this was encouraged by bounties, originally paid by soldiers' aid societies, then local governments, and finally in October 1863 by the federal government, offering a $300 bounty for volunteers and re-enlistments. Something like half a billion dollars were spent in bounties, and the savvy volunteer could honestly work the system (as localities and districts got into bounty bidding wars to fill their quotas) to get up to $1,000, or could dishonestly volunteer, collect a bounty, desert, volunteer again under a different name or in a different location, etc. You could even hire a "bounty broker" to help you find the best enlistment bounty option available (for a fee, of course)!. Similarly there were "substitute brokers" for the market in substitutes for draftees.