Spring (March to May) and autumn (October to November) are the most pleasant — warm, dry, with the gardens at their best. Summer is genuinely hot, with daytime highs over 35°C from June through September; the outdoor museums and Water Gardens are still doable in the early morning and evening.
The Fort Worth Stock Show and Rodeo runs the second half of January into early February and is the city's biggest annual event — book accommodation well in advance. The Main Street Arts Festival in April closes downtown for a long weekend and is worth timing a visit around.