The word we are looking for is:
COMPLETION!
Each list is an anagrammed set of words where each word is missing one letter. The missing letters from each word spell out another member of the set.
For example, ACENT + S anagrams to SECANT.
ACENT / CENOS / AEGNTT / ACGNNOTT
Trigonometry:
SECANT / COSINE / TANGENT / COTANGENT > SINE
EEEJNSWY / AEEHIMNPRSW / AACHILNNORRT / AAEERLW / KNORWY / AGIINRV / AACEHMSSSTTU / AAEINNNPSVY / EGGIOR / CCCEINOTTU / AALMNRY
13 Colonies:
NEWJERSEY / NEWHAMPSHIRE / NORTHCAROLINA / DELAWARE / NEWYORK / VIRGINIA / MASSACHUSETTS / PENNSYLVANIA / GEORGIA / CONNECTICUT / MARYLAND > RHODE ISLAND
AACDEEHLLOOPRRSST / ACT / GMPSY / CDMNRUU / AQTUZ / ACCELT / AOPZ / AAIPTT
Moh's scale
ORTHOCLASE FELDSPAR / TALC / GYPSUM / CORUNDUM / QUARTZ / CALCITE / TOPAZ / APATITE > FLUORITE
AHLLOW / CEERSS / EHINPX / AAABKZ / FIR
Harry Potter Title Endings
HALLOWS / SECRETS / PHOENIX / AZKABAN / FIRE > STONE
AIINU / HOU / AAEHKLOO / IMU / AAKU / AKLMOO
Hawaiian Islands
NIIHAU / OAHU / KAHAOOLAWE / MAUI / KAUAI MOLOKAI > HAWAII
DNOR / KNOPTY / AENNOOSS / ENNX / ENO
Noble elements:
RADON/ KRYPTON / OGANESSON / XENON / NEON > ARGON
AMR / EHRT / EIJPRU / CEMRRY / ANSUU / ESUV
Planets:
MARS / EARTH / JUPITER / MERCURY / URANUS / VENUS > SATURN
EIMSSW / DFOO / EGLLOS / AADFLN / BIMORR / EMRY / IIPPP
Fellowship of the Ring:
SAMWISE / FRODO / LEGOLAS / GANDALF / BOROMIR / MERRY/ PIPPIN > ARAGORN
AENOR / DE / ILOTV / BLU / DGIIO
Colors of the rainbow:
ORANGE / RED / VIOLET / BLUE / INDIGO > GREEN
AEHRS / CENPRR / EIVX / DIPU / CMOT / DENNO
Reindeer:
DASHER / PRANCER / VIXEN / CUPID / COMET / DONNER > PRANCER
In all, we have:
SINE, RHODEISLAND, FLUORITE, STONE, HAWAII, ARGON, SATURN, ARAGORN, GREEN, PRANCER
Applying the list of numbers as indexes into the words, we get:
(3,3,7,2,1,1,3,5,5,5)
N O T T H A T O N E
(1,2,4,1,1,2,4,2,5,7)
S H O N H R U R N R
The bottom doesn’t make sense; let’s see what was missing from each list:
COSECANT, SOUTHCAROLINA, DIAMOND, PRINCE, LANAI, HELIUM, NEPTUNE, GIMLI, YELLOW, BLITZEN
(1,2,4,1,1,2,4,2,5,7)
C O M P L E T I O N
Since the answer is a word, it’s not that (first) one; it’s COMPLETION.
Striving for excellence? Perfection? Success? Each list must be finished, so you are striving for completion!