Groebner bases are computed with the
gb command; see
gb. It returns an object of class
GroebnerBasis.
i1 : R = ZZ/1277[x,y];

i2 : I = ideal(x^3  2*x*y, x^2*y  2*y^2 + x);
o2 : Ideal of R

i3 : g = gb I
o3 = GroebnerBasis[status: done; Spairs encountered up to degree 8]
o3 : GroebnerBasis

To get the polynomials in the Groebner basis, use
gens
i4 : gens g
o4 =  y2+638x xy x2 
1 3
o4 : Matrix R < R

How do we control the computation of Groebner bases? If we are working with homogeneous ideals, we may stop the computation of a Groebner basis after Spolynomials up to a certain degree have been handled, with the option
DegreeLimit. (This is meaningful only in homogeneous cases.)
i5 : R = ZZ/1277[x,y,z,w];

i6 : I = ideal(x*yz^2,y^2w^2);
o6 : Ideal of R

i7 : g2 = gb(I,DegreeLimit => 2)
o7 = GroebnerBasis[status: DegreeLimit; all Spairs handled up to degree 2]
o7 : GroebnerBasis

i8 : gens g2
o8 =  y2w2 xyz2 
1 2
o8 : Matrix R < R

The result of the computation is stored internally, so when
gb is called with a higher degree limit, only the additionally required computation is done.
i9 : g3 = gb(I,DegreeLimit => 3);

i10 : gens g3
o10 =  y2w2 xyz2 yz2xw2 
1 3
o10 : Matrix R < R

The second computation advances the state of the Groebner basis object started by the first, and the two results are exactly the same Groebner basis object.
i11 : g2
o11 = GroebnerBasis[status: DegreeLimit; all Spairs handled up to degree 3]
o11 : GroebnerBasis

i12 : g2 === g3
o12 = true

The option
PairLimit can be used to stop after a certain number of Spolynomials have been reduced. After being reduced, the Spolynomial is added to the basis, or a syzygy has been found.
i13 : I = ideal(x*yz^2,y^2w^2)
2 2 2
o13 = ideal (x*y  z , y  w )
o13 : Ideal of R

i14 : gb(I,PairLimit => 2)
o14 = GroebnerBasis[status: PairLimit; all Spairs handled up to degree 1]
o14 : GroebnerBasis

i15 : gb(I,PairLimit => 3)
o15 = GroebnerBasis[status: PairLimit; all Spairs handled up to degree 2]
o15 : GroebnerBasis

The option
BasisElementLimit can be used to stop after a certain number of basis elements have been found.
i16 : I = ideal(x*yz^2,y^2w^2)
2 2 2
o16 = ideal (x*y  z , y  w )
o16 : Ideal of R

i17 : gb(I,BasisElementLimit => 2)
o17 = GroebnerBasis[status: BasisElementLimit; all Spairs handled up to degree 1]
o17 : GroebnerBasis

i18 : gb(I,BasisElementLimit => 3)
o18 = GroebnerBasis[status: BasisElementLimit; all Spairs handled up to degree 2]
o18 : GroebnerBasis

The option
CodimensionLimit can be used to stop after the apparent codimension, as gauged by the leading terms of the basis elements found so far, reaches a certain number.
The option
SubringLimit can be used to stop after a certain number of basis elements in a subring have been found. The subring is determined by the monomial ordering in use. For
Eliminate n the subring consists of those polynomials not involving any of the first
n variables. For
Lex the subring consists of those polynomials not involving the first variable. For
ProductOrder {m,n,p} the subring consists of those polynomials not involving the first
m variables.
Here is an example where we are satisfied to find one relation from which the variable
t has been eliminated.
i19 : R = ZZ/1277[t,F,G,MonomialOrder => Eliminate 1];

i20 : I = ideal(F  (t^3 + t^2 + 1), G  (t^4  t))
3 2 4
o20 = ideal ( t  t + F  1,  t + t + G)
o20 : Ideal of R

i21 : transpose gens gb (I, SubringLimit => 1)
o21 = {4}  F47F32F2G4FG2G3+18F2+3FG+6G221FG+9 
{3}  tG2tF+6tG+5tF3+3F2+3FG+3G2+G2 
{3}  tFG+tF4tG3t+F2FGG24FG+3 
{3}  tF24tF+tG+5tF2FG+3F+3G2 
{2}  t2+tF2tFG+1 
5 1
o21 : Matrix R < R

Sometimes a Groebner basis computation can seem to last forever. An ongoing visual display of its progress can be obtained with
gbTrace.
i22 : gbTrace = 3
o22 = 3

i23 : I = ideal(x*yz^2,y^2w^2)
2 2 2
o23 = ideal (x*y  z , y  w )
ZZ
o23 : Ideal of [x..z, w]
1277

i24 : gb I
 registering gb 5 at 0x7c95d2cf2a80
 [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
 number of monomials = 8
 #reduction steps = 2
 #spairs done = 6
 ncalls = 0
 nloop = 0
 nsaved = 0

o24 = GroebnerBasis[status: done; Spairs encountered up to degree 4]
o24 : GroebnerBasis

Here is what the tracing symbols indicate.
{2} ready to reduce Spolynomials of degree 2
(0) there are 0 more Spolynomials (the basis is empty)
g the generator yxz2 has been added to the basis
g the generator y2w2 has been added to the basis
{3} ready to reduce Spolynomials of degree 3
(1) there is 1 more Spolynomial
m the reduced Spolynomial yz2xw2 has been added to the basis
{4} ready to reduce Spolynomials of degree 4
(2) there are 2 more Spolynomials
m the reduced Spolynomial z4x2w2 has been added to the basis
o an Spolynomial reduced to zero and has been discarded
{5} ready to reduce Spolynomials of degree 5
(1) there is 1 more Spolynomial
o an Spolynomial reduced to zero and has been discarded
Let's turn off the tracing.
i25 : gbTrace = 0
o25 = 0

Each of the operations dealing with Groebner bases may be interrupted or stopped (by typing CTRLC). The computation is continued by reissuing the same command. Alternatively, the command can be issued with the option
StopBeforeComputation => true. It will stop immediately, and return a Groebner basis object that can be inspected with
gens or
syz. The computation can be continued later.
i26 : R = ZZ/1277[x..z];

i27 : I = ideal(x*y+y*z, y^2, x^2);
o27 : Ideal of R

i28 : g = gb(I, StopBeforeComputation => true)
o28 = GroebnerBasis[status: not started; all Spairs handled up to degree 1]
o28 : GroebnerBasis

i29 : gens g
o29 = 0
1
o29 : Matrix R < 0

The function
forceGB can be used to create a Groebner basis object with a specified Groebner basis. No computation is performed to check whether the specified basis is a Groebner basis.
If the Poincare polynomial (or Hilbert function) for a homogeneous submodule
M is known, you can speed up the computation of a Groebner basis by informing the system. This is done by storing the Poincare polynomial in
M.cache.poincare.
As an example, we compute the Groebner basis of a random ideal, which is almost certainly a complete intersection, in which case we know the Hilbert function already.
i30 : R = ZZ/1277[a..e];

i31 : T = (degreesRing R)_0
o31 = T
o31 : ZZ[T]

i32 : f = random(R^1,R^{3,3,5,6});
1 4
o32 : Matrix R < R

i33 : time betti gb f
 used 0.24386 seconds
0 1
o33 = total: 1 53
0: 1 .
1: . .
2: . 2
3: . 1
4: . 2
5: . 3
6: . 5
7: . 5
8: . 8
9: . 9
10: . 8
11: . 6
12: . 3
13: . 1
o33 : BettiTally

The matrix was randomly chosen, and we'd like to use the same one again, but this time with a hint about the Hilbert function, so first we must erase the memory of the Groebner basis computed above.
i34 : remove(f.cache,{false,0})

Now we provide the hint and compute the Groebner basis anew.
i35 : poincare cokernel f = (1T^3)*(1T^3)*(1T^5)*(1T^6)  cache poincare
3 5 8 9 12 14 17
o35 = 1  2T  T + 2T + 2T  T  2T + T
o35 : ZZ[T]

i36 : time betti gb f
 used 0.00342635 seconds
0 1
o36 = total: 1 53
0: 1 .
1: . .
2: . 2
3: . 1
4: . 2
5: . 3
6: . 5
7: . 5
8: . 8
9: . 9
10: . 8
11: . 6
12: . 3
13: . 1
o36 : BettiTally

The computation turns out to be substantially faster.