Piercings were covered here.
Tattoos are a huge concern for converts and baalei teshuva. Before becoming observant, they may have gotten tattoos, and now Jews have filled them with all kinds of unfounded fears. You will not be denied burial in a Jewish cemetery. You may use the mikvah. You might not have to get the tattoos removed.
Of course, at least with tattoo removal, you should consult your rabbi. If it's a tattoo of another religion, you will very likely be asked to remove it. However, there are issues if the removal will cause more physical harm than leaving it. (Sometimes removal procedures can result in significant scarring.) In order to prevent any further "damage" to the body, some rabbis do not recommend removal unless otherwise necessary or desired.
Keep in mind that there is a third option if removal of an inappropriate tattoo isn't possible for some reason: cover ups. If you aren't familiar with this term, it's using an old, unwanted tattoo as the basis for a new tattoo: "hiding" the old one. However, you will want a very good artist in order to avoid a catastrophe of a new tattoo that doesn't even hide the old one. Ask for references and examples.
Now where does this issue come from? Tattoos are still relatively taboo in the Jewish world, even in the more liberal movements. The younger generations are getting them increasingly, but even before you get to the halachic issues, there is a serious stigma from the Holocaust. Just in case you aren't already aware, the Nazis tattooed concentration camp inmates with a serial number on their forearms, and to the Nazis, that was their identification. To this day, there is a serious emotional reaction to tattoos in the entire Jewish community.
What are the halachic issues? a) Like piercings, tattoos are changing the natural form of your body. Your body is a temple, after all! b) In Biblical times, tattoos were used to show allegiance to particular idols. c) There is a specific Torah commandment against self-injury during mourning, which were also idolatrous practices. "You shall not make any cuttings in your flesh for the dead, nor imprint any marks upon you: I am HaShem." (Vayikra/Leviticus 19:28). This includes purposeful self-mutilation, cutting, scarification, burning, etc. Note that the psychological self-mutilation issue is very different from these kinds of actions being done for the sake of beauty or idolatry. Of course, the alternative argument is that this isn't done for the purpose of mourning or idolatry. Rabbis will still often rule against tattooing.
And just like piercings, the other major issue is fitting in to orthodox society. If you have a visible tattoo, you may want to consider having it removed simply for peace's sake (or your shidduch chances!). Of course, it may matter where that visible tattoo is. The arms or legs are probably different very different from a facial tattoo!
In almost any community, you will not be the only tattooed individual going to the mikvah. They've seen it before, and you'll be ok. There is no prohibition against you using the mikvah unless the community has made one. I'm sure there are places that would deny you mikvah services for that. However, it may matter to explain to the community rabbi in advance that you have tattoos, but they were done before your conversion. If that satisfies his concern for community standards, he can give an approval for you to use the mikvah.
Now to the big deal: there is no halacha that says tattooed folks can't be buried in a Jewish cemetery! Some uninformed people go so far as to tell these tattooed people that they have to have their tattooed arm cut off and buried elsewhere! That is simply not true. Further, that kind of postmortem dismemberment is blatantly against halacha as disrespecting the dead. It is possible that individual cemeteries or burial societies (the chevra kadisha) may have policies against it. Gd-willing, you will have the time to purchase your own burial plot and will be aware of any policy long before it's necessary. Also, as the times have changed, these societies may be forced to change their policies rather than deny burial to a growing percentage of tattooed Jews. (As a sidenote, since the argument is that a "voluntary" tattoo is against halacha, it may matter that the tattoo was done pre-conversion, and thus, not prohibited at the time.)
So that you don't have to believe me on the burial issue, here are three respectable sources that say the same:
The Orthodox Union: Very detailed and footnoted.
Timing: Whatever you decide, you need to have it done before your conversion. After the mikvah, you will be held to the same halacha, and you will probably be stuck with what you've got. Think about that carefully.