All translations are biased. Some translations have their bias obvious (which is good) or aimed at a target that is no longer applicable (also good); others are subtly problematic.
Avoid, at all costs, any translation that goes nuts with paraphrasing (moderate paraphrasing is usually okay). Mostly this means The Message. Also avoid any translation that is only used by a single denomination; semi-related to this, I also suggest avoiding any "study Bible", which inevitably consists more of third-party ideas telling you want to think about the passage rather than just reading it for yourself. I have nothing, however, against mere narrow-margin cross-references and rare single-word translation notes.
Other than that, for the gospels it doesn't matter much. Outside the gospels the differences become much more apparent.
The KJV (sometimes KJB or AV) has been highly influential on the English language (and, per its preface, the translators were very aware it would be). If you can stand the "thee"s and "thou"s (which do add important pluralization meaning in some places) and archaic spellings, it remains of excellent quality in most passages - better, even, than many modern translations. Its biases mostly fail to take hold of a modern audience. However, it does measurably suffer from the lack of modern linguistic scholarship in some passages (mostly, idiom-like things in the Old Testament, especially Job/Psalms/Proverbs and the prophets). Traditional editions do lack the quotation marks and paragraph flow that modern translations tend to use (though basic paragraph marks are usually present). Note that what people actually use is usually the 1769 edition, not (as some only-ists claim, the "original" 1611 which is unreadable), though there are some minor variations (such as "graffed" vs "grafted").
If you want something that keeps the KJV's advantages but is more modern, the ASV is mostly obsolete, but the NKJV is recent and pretty common (at the cost of giving up on the pluralization disambiguations). The KJ21, though much rarer, is quite useful because it does preserve the pluralization but modernizes the rest of the wording.
The NIV is popular, but I actually find it quite problematic. I have caught it on several occasions flat-out making something up, far beyond what can be explained by being a moderate paraphrase or even preferring different manuscripts. In particular, if someone in a debate cites the NIV, they're probably wrong. I think I've only once found a verse where the NIV gave a useful reading where other translations did not.
The ESV is not bad. It has its (unavoidable) biases but they are mostly of the obvious sort despite being aimed at a modern audience. However, I have found a few places where it introduces errors and ambiguities not in the KJV, such as Luke 22:31-32 (rigor summary for this verse: KJV/KJ21/NRSV/NASB good, NKJV bad as usual, NIV good for once, ESV bad for once). It is also prone to the problem of there being different editions in the wild with non-obvious differences.
The NRSV and NASB also seem relatively widely regarded, though I'm less familiar with them despite using them as parallels. They do succeed at the Luke 22 rigor test, at least in the versions I happen to have accessible (unfortunately, like the ESV, they do have non-obvious version differences).
There are a handful of other decent translations not mentioned here. But there are also a lot of low-quality translations, as well as minor variants of existing translations.
The YLT (not the Mormon "Young") is obscure but useful if you don't know enough Greek/Hebrew to efficiently use an interlinear directly, because it goes all in on literal translation, even for verb tenses that are unconventional in English.
A brief note on Bible translations.
All translations are biased. Some translations have their bias obvious (which is good) or aimed at a target that is no longer applicable (also good); others are subtly problematic.
Avoid, at all costs, any translation that goes nuts with paraphrasing (moderate paraphrasing is usually okay). Mostly this means The Message. Also avoid any translation that is only used by a single denomination; semi-related to this, I also suggest avoiding any "study Bible", which inevitably consists more of third-party ideas telling you want to think about the passage rather than just reading it for yourself. I have nothing, however, against mere narrow-margin cross-references and rare single-word translation notes.
Other than that, for the gospels it doesn't matter much. Outside the gospels the differences become much more apparent.
The KJV (sometimes KJB or AV) has been highly influential on the English language (and, per its preface, the translators were very aware it would be). If you can stand the "thee"s and "thou"s (which do add important pluralization meaning in some places) and archaic spellings, it remains of excellent quality in most passages - better, even, than many modern translations. Its biases mostly fail to take hold of a modern audience. However, it does measurably suffer from the lack of modern linguistic scholarship in some passages (mostly, idiom-like things in the Old Testament, especially Job/Psalms/Proverbs and the prophets). Traditional editions do lack the quotation marks and paragraph flow that modern translations tend to use (though basic paragraph marks are usually present). Note that what people actually use is usually the 1769 edition, not (as some only-ists claim, the "original" 1611 which is unreadable), though there are some minor variations (such as "graffed" vs "grafted").
If you want something that keeps the KJV's advantages but is more modern, the ASV is mostly obsolete, but the NKJV is recent and pretty common (at the cost of giving up on the pluralization disambiguations). The KJ21, though much rarer, is quite useful because it does preserve the pluralization but modernizes the rest of the wording.
The NIV is popular, but I actually find it quite problematic. I have caught it on several occasions flat-out making something up, far beyond what can be explained by being a moderate paraphrase or even preferring different manuscripts. In particular, if someone in a debate cites the NIV, they're probably wrong. I think I've only once found a verse where the NIV gave a useful reading where other translations did not.
The ESV is not bad. It has its (unavoidable) biases but they are mostly of the obvious sort despite being aimed at a modern audience. However, I have found a few places where it introduces errors and ambiguities not in the KJV, such as Luke 22:31-32 (rigor summary for this verse: KJV/KJ21/NRSV/NASB good, NKJV bad as usual, NIV good for once, ESV bad for once). It is also prone to the problem of there being different editions in the wild with non-obvious differences.
The NRSV and NASB also seem relatively widely regarded, though I'm less familiar with them despite using them as parallels. They do succeed at the Luke 22 rigor test, at least in the versions I happen to have accessible (unfortunately, like the ESV, they do have non-obvious version differences).
There are a handful of other decent translations not mentioned here. But there are also a lot of low-quality translations, as well as minor variants of existing translations.
The YLT (not the Mormon "Young") is obscure but useful if you don't know enough Greek/Hebrew to efficiently use an interlinear directly, because it goes all in on literal translation, even for verb tenses that are unconventional in English.