The goal of the proposed project is to combine the color, flavor, and production of "Lemon" with the vining habit, vigor, and insect resistance of "Tatume". They were seeking input on what strategy to pursue with the F1 plants; backcrossing to "Tatume" or selfing.
Backcrossing is generally used to transfer a limited number of traits from one genetic background to another, without dragging along unrelated genetic baggage. This technique was used to transfer anthocyanin pigment production from wild tomato relatives (Solanum cheesmanii & S. chilense) into the garden tomato (S. lysopersicum var. "Indigo Rose") . It was also used to transfer the red factor from the Pine Siskin (Carduellis cucullata) into Canary birds (Serinus canaria domestica) . In both cases, there were many other traits of the donor species which were not desired in the final product and that were successfully filtered out via backcrossing.
The genetics of squash color is relatively well worked out: two loci interact to produce green (wwgg), yellow (wwG_), or white (W_G_) immature fruit . The genetics of vining habit is also known: one locus produces bush (Bu_) or trailing (bubu) vine habit . The genetics of flavor, yield, and many other traits are less well understood.
If you don't know the genetics of the traits you're interested in, you should first grow out a reasonable number of F2 progeny in order to characterize the inheritance patterns. Growing 20 or so F2s from a selfed-F1 might not give you every expected combination of traits, but it will allow you to examine the ratios of each of the traits you're interested in. The recessive version of each trait will be in the minority.
Backcrossing has specific consequences to a breeding program depending on the genetics of the trait(s) you're looking to transfer.
"Tatume" x "Lemon"
P1 : (wwgg) green x (wwGG) yellow
F1 : (wwGg) yellow
"Tatume" x F1
P2 : (wwgg) green x (wwGg) yellow
BC1 : (wwGg) yellow + (wwgg) green
F2 : 3 (1 BiBi; 2 Bibi) bitter + 1 (bibi) sweet
"Tatume" x F1
P2 : (BiBi) bitter x (Bibi) bitter
BC1F1 : (1 BiBi; 1 Bibi) bitter
"Tatume" x F2
P2 : (BiBi) bitter x (bibi) bitter
BC1F1 : (Bibi) bitter
In the dominant case, it is apparent which of the progeny carry the trait of interest. You can immediately choose which plants to use in the next stage.
In the recessive case, it is not apparent which of the backcross progeny carry the trait of interest. You will have a 50% chance of losing the recessive allele you're interested in during every backcross generation. The solution to this is to screen a set of F2s in every cycle to find progeny carrying two copies of the recessive allele.
"Tatume" x BC1
P3 : (wwgg) green x (wwGg) yellow
BC2 : (wwGg) yellow + (wwgg) green
P3 : (BiBi) bitter x (bibi) sweet
BC2F1 : (Bibi) bitter
BC2F2 : 3 (1 BiBi; 2 Bibi) bitter + 1 (bibi) sweet
After several cycles of recurrent backcrossing with a dominant trait, the final progeny will be very much like the backcross parent ("Tatume"), but with the exception that they will have both dominant and recessive alleles for the selected locus. A round of selfing to produce F2s, then another round of selfing to produce F2 families, then selecting those F2 families which did not produce any green squash will get rid of the undesired recessive allele from the "Tatume" parent line.
After several cycles of recurrent backcrossing with a recessive trait, the final progeny will be very much like the backcross parent ("Tatume"), with the exception that they will have two recessive alleles at the selected locus. No further selection will be required to stabilize the line, but it will have taken twice as many years to get here compared to the case of transferring a dominant trait.
If you instead screen large numbers of F2s from the original "Tatume" x "Lemon" cross, you might find the combination of dominant and recessive traits you're looking for in the second year. The recessive traits will already be stabilized, while the dominant traits will require more work. Self each F2 plant and save the seeds separately to make F2 families. Growing out the F2 families will let you see which F2 plants had a recessive allele hiding under the desired dominant trait. In addition, this method will let you examine a large number of combinations of genes you weren't expecting in advance.
If one parent was poisonous or had many other negative traits, you would get very few edible progeny and your time would probably be better spent using a backcrossing strategy to filter out everything except the few traits you're interested in transferring.
Both parents in the proposed project are highly edible and only differ in a few charismatic traits, so the vast majority of F2 progeny will be highly edible and have some combination of those traits. In this case, the potential to uncover interesting combinations of hidden alleles out-ranks the limited potential to produce inedible progeny.