I'll start with some basic things that are important to know.
In Denmark you will find that almost everyone used a system of patronymics, which means that a person's surname is based on their father's name, with -sen or -datter added to the name. So if a person's name is Hans, his sons' surname will be Hansen and his daughters' surname will be Hansdatter.
Around the mid-1800s, there was a transition to using -sen for their surname, regardless of gender. Many women who were christened with -datter continued to use the suffix, while others changed to -sen, but in general the patronymic system continued.
In the late 1800s, for most families the surnames began to be fixed. For example, the children of my ancestor, Jens Peter Pedersen, all had the surname Jensen. But his grandchildren had the surname Jensen regardless of the father's name. In addition, many people began changing their surnames to names that did not have the -sen suffix, often with the surname reflecting their residence or another characteristic about them. For example, Jens Peter Pedersen's son, Jens Christian Jensen, inherited the family farm, which was named Lille Kousholt, and around 1910 all of his children's surnames were changed to Kousholt. Often this name change will be noted in the person's baptism record.
My observation has been that women generally kept their patronymic surname up until the early 1900s, and then an increasing number of women started taking their husband's surname.
Same Name is Not Enough
In the 1800s and earlier, there was not a lot of variety in names. So if you find a record with the right name and date you can't just assume it's the same person. Even within the same Sogn (parish) you will often find multiple people with the same name born in the same year. So when you are looking at records you should look for other clues in addition to the name:
- Relationships: Research entire families at the same time rather than focusing on individuals. If you know the names and dates of the parents and siblings it is easier to tell if you have found the right person in a census record because you can compare the entire family to what you already know. In parish records check the names and homes of the witnesses or godparents because often these will be family members. If you are reading a marriage record and you recognize one of the witnesses' names as the father, or in a christening record you recognize names of grandparents or aunts or uncles, then you can be more confident that the record is for the person you are researching.
- Location: Whenever possible you should know location all the way down to the village or farm where somebody lives. Continuing with the example of Jens Christian Jensen, reading through parish records I can find many parents with that same name, but I can quickly tell that they are not the same person because their residence is not Lille Kousholt. People did move around some, but generally they didn't move much farther than adjacent parishes, which would be less than 15 miles. So if you are researching someone who lived in northern Denmark (Hjorring), it's extremely unlikely that a record from Copenhagen (100 miles away, and across a body of water) is about the same person.
In Denmark places are generally broken down like this (from biggest to smallest): Country (Denmark), Amt (County), Herred (Hundred), Sogn (Parish), By (City/Town), and sometimes the farm name/house number/street. When I am entering places on Family Search I try to include each of these as much as possible.
To get a better sense of the geography I recommend playing with the Denmark clickable map on FamilySearch.org's Denmark Genealogy Research Wiki. Clicking on the Amt (county) will take you to that wiki page, which will generally have another clickable map showing the parishes with borders around the herreds. Many parishes' wiki pages also have a list of placenames within the parish. I actually highly recommend reading everything on the research wiki about Denmark Genealogy because there is a lot of useful information there.
The most useful records you will find in Denmark are parish records and census records. Parish records can tell you names, birth/death/marriage dates and places, parents, and other relatives. Census records are very valuable for finding whole families and giving you a general year and place of birth.
Census records are partially indexed and can be found on MyHeritage.com (click Nordic census on the right side) or danishfamilysearch.com. Danish Family Search is useful because you can either do a general search by adding the basic information, or you can search within specific parishes by first name, surname, or placename. On Danish Family Search, many censuses are indexed but for some reason don't get included in searches, so you have to find them by going to that parish's census list. The censuses available on MyHeritage all seem to be indexed and searchable, but sometimes it can be hard to find a person in a search if their name was indexed incorrectly.
Many (not all) Danish parish records were indexed by the LDS church and the indexes are available on familysearch.org. This can be very useful, but it's important to look up the image because it's actually not that uncommon for there to be two couples in the same parish with the same names, so you will need to look up the name of their farm or village.
Many Danish parish records are also indexed with images on myheritage.com.
Images of Danish parish records are also available on danishfamilysearch.org and Arkivalieronline.com. I feel that the interface of Arkivalieronline seems to be more usable for browsing through images, but both are good and seem to have pretty much all the available parish records. I use these two websites constantly when I am doing Danish family history.
Recently I uploaded my family tree from RootsMagic onto MyHeritage.com to see if I could get any record hints. Looking through the record hints they have given, I am very excited because there are many record hints that give me information about people I was stuck on. This feature will be an enormous help for my research. MyHeritage has many census and parish records indexed and mapped to the images.
Danish records in the 1800s and earlier were written in Gothic handwriting, which is similar to the cursive handwriting we are used to, but has many differences. When I first tried reading Danish parish records I almost gave up because the handwriting was so difficult to decipher.
The Family Search Research Wiki has a page with links about reading Gothic handwriting, which was very helpful to me. I would also suggest starting with people whose birth date and place you know, then find their birth records and try to read them. Since you already have a general idea of what the records say, that will help you start to get a sense of what the letters look like. Also keep in mind that there wasn't a lot of variety in names, so look for familiar names (Jens, Hans, Christian/Kristian, Soren, Anders, Margrethe, Karen, Maren, Mette, Kristine, Ane, Marie, etc.) to get a sense of what the letters look like.
It takes a while to get used to the handwriting, but the best way is to keep practicing.
Obviously if you don't know Danish then the language will be a challenge. On the Denmark Genealogy page of the Research Wiki, there are word lists that have common words that you would see on records, which is really useful. Google Translate is also a very helpful tool. You don't have to be fluent in Danish to do Danish family history. Learning the words that are most commonly used on the records and using the word lists and Google Translate will get you pretty far.