diff --git a/pandas/_libs/tslib.pyx b/pandas/_libs/tslib.pyx
index ee3964b892e2e..39c8aab2ce5dc 100644
--- a/pandas/_libs/tslib.pyx
+++ b/pandas/_libs/tslib.pyx
@@ -595,7 +595,7 @@ cpdef array_to_datetime(
                                 continue
                             elif is_raise:
                                 raise ValueError(
-                                    f"time data {val} doesn't match format specified"
+                                    f"time data \"{val}\" at position {i} doesn't match format specified"
                                 )
                             return values, tz_out
 
@@ -607,11 +607,11 @@ cpdef array_to_datetime(
                             # to check if all arguments have the same tzinfo
                             tz = py_dt.utcoffset()
 
-                        except (ValueError, OverflowError):
+                        except (ValueError, OverflowError) as err:
                             if is_coerce:
                                 iresult[i] = NPY_NAT
                                 continue
-                            raise TypeError("invalid string coercion to datetime")
+                            raise type(err)(f"invalid string coercion to datetime for \"{val}\" at position {i}")
 
                         if tz is not None:
                             seen_datetime_offset = True
@@ -664,11 +664,11 @@ cpdef array_to_datetime(
 
                         # Still raise OutOfBoundsDatetime,
                         # as error message is informative.
-                        raise
+                        raise OutOfBoundsDatetime(f"Cannot convert \"{val}\" at position {i} to datetime")
 
                     assert is_ignore
                     return values, tz_out
-                raise
+                raise OutOfBoundsDatetime(f"Cannot convert \"{val}\" at position {i} to datetime")
 
     except OutOfBoundsDatetime:
         if is_raise:
@@ -798,6 +798,7 @@ cdef _array_to_datetime_object(
     # We return an object array and only attempt to parse:
     # 1) NaT or NaT-like values
     # 2) datetime strings, which we return as datetime.datetime
+    # 3) special strings - "now" & "today"
     for i in range(n):
         val = values[i]
         if checknull_with_nat_and_na(val) or PyDateTime_Check(val):
@@ -811,17 +812,24 @@ cdef _array_to_datetime_object(
             if len(val) == 0 or val in nat_strings:
                 oresult[i] = 'NaT'
                 continue
+
             try:
-                oresult[i] = parse_datetime_string(val, dayfirst=dayfirst,
+                # Handling special case strings today & now
+                if val == "today":
+                    oresult[i] = datetime.today()
+                elif val == "now":
+                    oresult[i] = datetime.now()
+                else:
+                    oresult[i] = parse_datetime_string(val, dayfirst=dayfirst,
                                                    yearfirst=yearfirst)
-                pydatetime_to_dt64(oresult[i], &dts)
-                check_dts_bounds(&dts)
-            except (ValueError, OverflowError):
+                    pydatetime_to_dt64(oresult[i], &dts)
+                    check_dts_bounds(&dts)
+            except (ValueError, OverflowError) as err:
                 if is_coerce:
                     oresult[i] = <object>NaT
                     continue
                 if is_raise:
-                    raise
+                    raise type(err)(f"Unable to parse string \"{val}\" at position {i}")
                 return values, None
         else:
             if is_raise:
diff --git a/pandas/_libs/tslibs/parsing.pyx b/pandas/_libs/tslibs/parsing.pyx
index 5cb11436f6f45..50be534328334 100644
--- a/pandas/_libs/tslibs/parsing.pyx
+++ b/pandas/_libs/tslibs/parsing.pyx
@@ -261,7 +261,7 @@ def parse_datetime_string(
         datetime dt
 
     if not _does_string_look_like_datetime(date_string):
-        raise ValueError('Given date string not likely a datetime.')
+        raise ValueError(f'Given date string {date_string} not likely a datetime.')
 
     if does_string_look_like_time(date_string):
         # use current datetime as default, not pass _DEFAULT_DATETIME
@@ -287,7 +287,7 @@ def parse_datetime_string(
     except TypeError:
         # following may be raised from dateutil
         # TypeError: 'NoneType' object is not iterable
-        raise ValueError('Given date string not likely a datetime.')
+        raise ValueError(f'Given date string {date_string} not likely a datetime.')
 
     return dt
 
@@ -363,7 +363,7 @@ cdef parse_datetime_string_with_reso(
         int out_tzoffset
 
     if not _does_string_look_like_datetime(date_string):
-        raise ValueError('Given date string not likely a datetime.')
+        raise ValueError(f'Given date string {date_string} not likely a datetime.')
 
     parsed, reso = _parse_delimited_date(date_string, dayfirst)
     if parsed is not None: