Beginner Cryptic #1
Across
1 - COR[PS]E
5 - ME[MEN]TO (TOME anag.)
7 - L+EAT+HER
10 - F+RIGID

Down
1 - CO+(i)MPLY
2 - RAM (double def.)
3 - (ala)S I N(ever)
4 - POURED (RODE UP anag.)
6 - mEATy
8 - AIR (double def.)
9 - HUG(e)

Beginner Cryptic #2
Across
1 - PRET[EE]N (P(a)RENT anag.)
6 - R(e)A(d)Y
7 - TRADERS (S+RE+DART reversed)
8 - TRE[M]BLE
12 - MAY (double def.)
13 - PRUDENT (UP TREND anag.) 

Down
1 - PIT STOP (POTS TIP reversed)
2 - (long)ER A(ge) & lit.
3 - T[A]D (T(ed) D(anson) initially)
4 - EYE (double def.)
5 - NO(i)SIEST
9 - (f)EMU(r)
10 - MAD(e)
11 - B[Y]E (at first Y(ou'd))